u.version2.ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormaltextwarnerrorlistitemulletfunctgospalgorithmindefinitsummatusagcallsequencparameterexpressnamevariablsynopsidescriptimplementatdefinitalsohsumtheortheorycalculathypergeometricupwardantidifferenchypergeometrictermfgkgwhenevsuchexistprocedurreturntermggwithrespectrationaltypicalcaseratioproductrationafunctionpowerfactorialbinomialcoefficientpochhammsymbolinteglineartheirargumentimplementatiosupporttypegospermessagapplicabloumightwanttryextendnoprogramprovexamplreamplvpackaghypergeometricycopyrightwolframkoepfuniversitkasselgsiamreviewproblemkgffactorialgngbinomialgbinomialngfpochhammergterseealsocontgospsumrecurswzcertificat {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bul let Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } 0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 260 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" GVibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctclosedformsumrecurszeilbergalgorithmusagcallsequencparameterexpressnamesummatvariablrecurrencsynopsidescriptthesfunctionimplementatwhilecalculatupwardequatiosumsgngsumgfgkgagbghypergeometricwithrespectclosdformtrydeterminhypergeometrictermequalhypergeometrickgfrationalwittypicalcasratioproductpowerfactorialbinomialcoefficientpochhammsymbolinteglineartheirargumentsupporttypewithoutanyboundalwayhomogeneousassumfinitnaturalzeroutsidranginfinitsummationshouldusenthuenforcingcomputatinhomogeneoupartequaalsohsumtheorallprocedurespossiblobtaincertificatallowsimplposterioproofequatusingoptionalargumcertificattruewillreturnentrfirstcomputsecondcertificatnotelookonlyionfirstorderwhilsearchupmaxordlobalsetdefaultbutmaychangpositivevaluyoucanoptionalrecordnonnegatidexpand expansexpect expg# explicit explik  exponentialexpr  expresexpresst 1 "" {XPPMATH 20 "6#7$7#/,&-%\"fG6#,&%\"nG\"\"\"F, F,F,*&,&)%\"qG,&F+F,#F,\"\"#F,F,F,!\"\"F,-F(6#F+F,F,\"\"!7#/,&F'F,*&,& F/F4F,F4F,F5F,F,F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "re:= \+ q^2*q^(2*n)*S(n+3)+(1+q)*q^n*S(n+2)+\n(1-q^n)*S(n+1)-S(n)=0;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#reG/,**()%\"qG\"\"#\"\"\")F),$*&F*F+%\"nG F+F+F+-%\"SG6#,&F/F+\"\"$F+F+F+*(,&F+F+F)F+F+)F)F/F+-F16#,&F/F+F*F+F+F +*&,&F+F+F7!\"\"F+-F16#,&F/F+F+F+F+F+-F16#F/F=\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "qrecsolve(re,q,S(n),solution=series);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7#-%$SumG6$**))%\"qG%\"nG%#_iG\"\"\")F **$)F,\"\"#F-F-%$_C1GF--%,qpochhammerG6%F*F*F,!\"\"/F,;\"\"!%)infinity G" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" }{TEXT 256 1 " \+ " }{HYPERLNK 17 "qsum" 2 "qsum" "" }{TEXT 259 2 ", " }{HYPERLNK 17 "qs umrecursion" 2 "qsumrecursion" "" }{TEXT 258 4 " " }}}}{MARK "0 0 0 " 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } T 1 0 21 "qrecsolve(re,q,f( n));" }}{PARA 12 ""FPS,FPSFormalPowerSeries,FPSHolonomicDE,FPS SimpleDE,FPS SimpleRE,FPSreferences,hsum setting,qsumstandardsum,FPS theory,hsum theory,qsum the summa tion bounds are " }{TEXT 258 11 "not natural" }{TEXT -1 162 ", one sho uld always use the call\n sumrecursion(f,k=a..b,s(n));\nthus enfo rcing the computation of the inhomogeneous part of the recurrence equa tion (see also " }{HYPERLNK 17 "hsum[theory]" 2 "hsum[theory]" "" } {TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXT -1 60 "With all these procedu res it is also possible to obtain the " }{TEXT 261 20 "rational certif icate" }{TEXT -1 6 " (see " }{HYPERLNK 17 "hsum[theory]" 2 "hsum[theor y]" "" }{TEXT -1 239 ") which allows a simple a posterio proof of the \+ calculated recurrence equation. When using the optional argument certi ficate=true, sumrecursion will return a list with two entries. The fir st is the computed recurrence equation for the sum " }{XPPEDIT 18 0 "s (n)" "6#-%\"sG6#%\"nG" }{TEXT -1 29 " and the second entry is the " } {TEXT 262 20 "rational certificate" }{TEXT -1 1 "." }}{PARA 15 "" 0 " " {TEXT -1 145 "Note that zeilberger looks only for a recurrence equat ion of first orderM2 ", " }{TEXT 19 16 "gausselim_normal" }{TEXT -1 98 " determines which procedure is used f or simplification during the process of Gaussian elimination." }} {PARA 15 "" 0 "" {TEXT -1 11 "The option " }{TEXT 19 8 "recorder" } {TEXT -1 57 " may be set to a posint or a range of posint, default is \+ " }{TEXT 19 4 "1..5" }{TEXT -1 62 ".\nIf a range is specified qsumdiff eq will start to look for a " }{TEXT 285 1 "q" }{TEXT -1 79 "-differen ce equation in that range. A positive integer n is equivalent to n..n. " }}{PARA 15 "" 0 "" {TEXT -1 11 "The option " }{TEXT 19 14 "evalqdiff =true" }{TEXT -1 25 " can be used to return a " }{XPPEDIT 18 0 "q" "6# %\"qG" }{TEXT -1 47 "-recurrence equation for the sum (instead of a " }{TEXT 287 1 "q" }{TEXT -1 22 "-difference equation)." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G" }}{PARA 12 "" 1 "" {XPPMATH 20 6#%\"xG-%$sumG6$-%\"fG6$F'%\"kG/F.;,$%)infinityG!\"\"%)infinityG" } {TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 43 "where the summation ran ge is assumed to be " }{TEXT 19 19 "-infinity..infinity" }{TEXT -1 1 " ." }}{PARA 15 "" 0 "" {TEXT -1 48 "If successful, qsumdiffeq returns a homogeneous " }{TEXT 284 1 "q" }{TEXT -1 24 "-difference equation in \+ " }{XPPEDIT 18 0 "s(x);" "6#-%\"sG6#%\"xG" }{TEXT -1 41 ", otherwise a n error message is returned." }}{PARA 15 "" 0 "" {TEXT -1 136 "The las t four calling sequences are useful shortcuts if you want to apply qsu mdiffeq to basic generalized hypergeometric functions (see " } {HYPERLNK 17 "qhyperterm" 2 "qhyperterm" "" }{TEXT -1 7 "). E.g." }} {PARA 257 "" 0 "" {TEXT 19 32 "qsumdiffeq(f,upper,lower,q,z,x);" }} {PARA 14 "" 0 "" {TEXT -1 17 "is a shortcut for" }}{PARA 257 "" 0 "" {TEXT 19 51 "qsumdiffeq(f*qhyperterm(upper,lower,qq,z,k),q,k,x);" } {TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 51 "If the prefactor f is e qual to one you can omit it." }}{PARA 15tingtiotion7tionaltmp tmpgtohyptopictorgtorialg toronto trailtreattric trieOtroltrue3#try# 7ibmintelntmathtimehyperlinkcommhelpheadnormaltimeslistitembulletintroductqsumpackagcopyrightharaldingwolframkoepfuniversitykasselcommentbugreportrewelcomsentsecondauthorviamailprofdruniversitndepartmematiccomputsciencnheinrichplettstrndmathematikunideurlhttpwwwdescriptqsumimplementatqganalogugospzeilbergshortdescriptiontheorpetkovsekalgorithmhebasedhypergeometriccasedescribsummatalgorithmicapproachspecialfunctidentitiviewegbraunschweigwiesbadenisbnproceduresintroducbookprovidmaplhsummplsomeprocdurealsoavailablsumtoolfunctionqbinomialqbracketqfactorialqgammaqpochhammqfunctqhypertermqpsihypertermqratiosimpcombqsimpcombqsimplifqgosposperqsumrecursqsumrecurionqsumdiffeqqrecsolvsumqhypmaycustomglobalvariablsettreferencabramovbronsteinpetkovsekpolynomialsolutionlinearoperatorequationleveltedissacacmpresnewyorkgasprahmanbasicseriencyclopediamathematicapplica"n" "6#%\"nG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 232 ". This is typically the cas e for ratios of products of rational functions, powers, factorials, bi nomial coefficients, and Pochhammer symbols that are integer-linear in their arguments. The implementation supports this type of input." }} {PARA 15 "" 0 "" {TEXT -1 84 "The output of sumrecursion (without any \+ summation bounds) or zeilberger is always a " }{TEXT 257 22 "homogeneo us recurrence" }{TEXT -1 26 ". I.e. it is assumed that " }{XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 49 " has finite support an d the summation bounds are " }{TEXT 259 7 "natural" }{TEXT -1 2 " (" } {XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 38 " is zero o utside the summation range):" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "Sum(f (n,k),k=a[n]..b[n])=Sum(f(n,k),k=-infinity..infinity)" "6#/-%$SumG6$-% \"fG6$%\"nG%\"kG/F+;&%\"aG6#F*&%\"bG6#F*-F%6$-F(6$F*F+/F+;,$%)infinity G!\"\"F;" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 28 "If 7ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpeadingnormaltimesheadlistitembulletfunctqsumdiffeqanaloguezeilbergalgorithmfinddifferencequatcallequencqsumdiffeqnqsumupperlowerqqdiffeqparameteralgebraicexpressnqnamenksummatvariablenxvariablnsnaboundnupplowergeneralhypergeometricnqqintegnzevaluatpointsequencequationoptionalargumentdescriptimplementatqgalogucalculatsuminfinitsgxgsumgfgkginfinitygrangeassumsuccessfulreturnhomogeneouotherwiserrormessaglasfourusefulshortcutyouwantapplqsumdiffeqbasicfunctionqhypertermprefactorqualcanomitoptionaalwaytypenamlefthandsideoptionrightcorrespondvalufollowpossiblsettinformathowchangstandardmostconsumpartconsistsolvsystemlineartheorptionsolvemethoddeterminprocedurusedsetautoabpgausselimdefaultnifspecifibuiltinprocedurgaussianeliminatirstpropospaulriesreferencusingbpabramovbronsteinpetkovsekapplieists" }}{PARA 14 " " 0 "" {TEXT -1 78 "then the program has proved that no hypergeometric term antidifference exists." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "rea d `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeo metric~Summation\",~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfram~Koepf,~University~of~KasselG" }}} {PARA 0 "" 0 "" {TEXT -1 38 "see (SIAM Review, 1994, Problem 94-2) " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "gosper((-1)^(k+1)*(4*k+1)*( 2*k)!/(k!*4^k*(2*k-1)*(k+1)!),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*2\"\"#\"\"\",&%\"kGF&F&F&F&)!\"\"F'F&-%*factorialG6#,$*&F%F&F(F&F&F& -F,6#F(F*)\"\"%F(F*,&*&F%F&F(F&F&F&F*F*-F,6#F'F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gosper(binomial(k,n),k);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*(,&%\"kG!\"\"%\"nG\"\"\"F),&F(F)F)F)F'-%)binomialG 6$F&F(F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "gosper(binomi al(n,k)/2^n-binY 0 "> " 0 "" {MPLTEXT 1 0 36 "fasenmyer(k*binomial(n,k),k,s(n),1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"nG\"\"\"-%\"sG6#,&F&F'F'F'F'F' *(\"\"#F'-F)6#F&F'F+F'!\"\"\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "gosper" 2 "gosper" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "closedform, sumrecursion and zeilberger" 2 "sumrecursion" "" } {TEXT -1 1 " " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 } -F,6$F*F /F)!\"\"*(F&F)F.F)-F,6$F*F0F)F4\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(a[1,1]=1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /,(*&-%\"FG6$,&%\"nG\"\"\"F+F+,&%\"kGF+F+F+F+F*F+F+*&F)F+-F'6$F*F,F+! \"\"*&F)F+-F'6$F*F-F+F1\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "C alculate a recurrence for the sum " }{XPPEDIT 18 0 "s(n)=sum(k*binomia l(n,k),k=-infinity..infinity)" "6#/-%\"sG6#%\"nG-%$sumG6$*&%\"kG\"\"\" -%)binomialG6$F'F,F-/F,;,$%)infinityG!\"\"F4" }{TEXT -1 1 ":" }}{PARA  " }{TEXT 257 3 "f, " }{TEXT 261 28 "i.e. an equation of the form" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "sum(sum(a[i,j]*F(n+j,k+i),j=1..nmax),i=1..kmax)=0" "6#/-%$sumG6$-F% 6$*&&%\"aG6$%\"iG%\"jG\"\"\"-%\"FG6$,&%\"nGF/F.F/,&%\"kGF/F-F/F//F.;F/ %%nmaxG/F-;F/%%kmaxG\"\"!" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 10 "where the " }{XPPEDIT 18 0 "a[i,j]" "6#&%\"aG6$%\"iG%\"jG" } {TEXT -1 22 " are polynomials wrt. " }{XPPEDIT 18 0 "n" "6#%\"nG" } {TEXT -1 13 " and free of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 13 " is returned." }}{PARA 15 "" 0 "" {TEXT -1 3 "If " }{TEXT 262 1 "F" } {TEXT -1 5 " and " }{TEXT 263 1 "a" }{TEXT -1 147 " are not specified \+ as arguments to kfreerec, then the resulting recurrence equation conta ins these variable names as local variables. All symbolic " }{XPPEDIT 18 0 "a[i,j]" "6#&%\"aG6$%\"iG%\"jG" }{TEXT -1 62 " in the resulting r ecurrence equation are arbitrary constants." }}{PARA 15 "" 0 "" {TEXT -1 74 "The procedure fasenmyer delivers an upward recurrence equation \+ {VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {RialG6#F(F0" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "pochhammer" 2 "pochhammer" "" }{TEXT -1 3 " , " } {HYPERLNK 17 "Sumtohyper" 2 "Sumtohyper" "" }}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } &-%(productG6$-%+pochhammerG6$&%\"UG6#%\"iG%\"kG/F -;\"\"\"%\"pGF1-F%6$-F(6$&%\"LG6#F-F./F-;F1%\"qG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "z^k/k!" "6#*&)%\"zG%\"kG\"\"\"-%*factorialG6#F&!\"\" " }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Ma ple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfr am~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "hyperterm([a,b],[c],z,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* ,-%+pochhammerG6$%\"aG%\"kG\"\"\"-F%6$%\"bGF(F)-F%6$%\"cGF(!\"\")%\"zG F(F)-%*factor{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {$ a name, the summation vari able\nx - a name, the " }{TEXT 278 1 "q" }{TEXT 279 20 "-difference equation" }{TEXT -1 29 " variable\ns - a name, the " }{TEXT 280 1 "q" }{TEXT 281 20 "-difference equation" }{TEXT -1 193 " function\na, \+ b - algebraic expressions, upper and lower summation bounds\nupper, lo wer - lists of algebraic expressions, the upper and the lower\n \+ parameters of the general " }{TEXT 268 1 "q" }{TEXT -1 160 "-hypergeometric function\nqq - a name or a name^integer\nz \+ - an algebraic expression, the evaluation point\n... - a sequence o f equations, optional arguments" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 42 "This function is an \+ implementation of the " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 51 "-an alogue of Zeilberger's algorithm, calculating a " }{TEXT 283 1 "q" } {TEXT -1 32 "-difference equation for the sum" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "s(x) = sum(f(x,k),k = -infinity .. infinity);" "6#/-%\} 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "List Item" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 3 3 1 0 1 0 2 2 14 5 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 114 "GFrecursion, GFdiffeq - Calculate a recurrence/differential equat ion for a function given by a generating function" }}{PARA 0 "" 0 "usa ge" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 55 "\n GFrecursion(F,a ,z,s(n));\n GFdiffeq(F,a,z,n,s(x));" }}{PARA 0 "" 0 "" {TEXT 26 11 " Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 6 "F,a - " }{TEXT -1 17 "an expression\n " }{TEXT 23 10 "n,s,x,z - " }{TEXT -1 6 "a name" }}} {SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 88 "These functions determine a recurrence/differenti al equation for a familiy of functions " }{XPPEDIT 18 0 "f[n](x)" "6#- &%\"fG6#%\"nG6#%\"xG" }{TEXT -1 35 ", given by the generating function " }{XPPEDIT 18 0 "F(z)" "6#-%\"FG6#%\"zG" }{TEXT -1 5 " with" }} {PARA 257 "" 0 "" {XPP%ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctkfreerecfasenmyalgorithmusagcallsequenckmaxnmaxparameterexpressamesummatvariablrecurrencnamenonnegatintegerorderwrtarbitrarconstantsynopsidescriptfunctionimplementatasenmyprocedurfreereccalculatfreeequathypergeometrictermformsumsumgagigjgfgngfkgfnmaxgkmaxgpolynomialngkgreturnspecifiargumentresultcontainstheslocalallsymbolicecurrencdeliverupwardinfinitinfinitsginitygcalculatviasummandsummovernoteassumfinitsupporthomogeneounmaxgfexamplreadhsummplvpackagycopyrightwolframkoepfuniversitkasselgbinomialbinomialgbinomiasubsalculatinfinitygbinomialseealsoalsogospclosedformsumrecurszeilberg " }{TEXT 257 3 "f, " }{TEXT 261 28 "i.e. an equation of the form" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "sum(sum(a[i,j]*F(n+j,k+i),j=1..nmax),i=1..kmax)=0" "6#/-%$sumG6$-F% 6$*&&%\"aG6$%\"iG%\"jG\"\"\"-%\"FG6$,&%\"nGF/F.F/,&%\"kGF/F-F/F//F.;F/ %%nmaxG/F-;F/%%kmaxG\"\"!" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 10 "where the " }{XPPEDIT 18 0 "a[i,j]" "6#&%\"aG6$%\"iG%\"jG" } {TEXT -1 22 " are polynomials wrt. " }{XPPEDIT 18 0 "n" "6#%\"nG" } {TEXT -1 13 " and free of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 13 " is returned." }}{PARA 15 "" 0 "" {TEXT -1 3 "If " }{TEXT 262 1 "F" } {TEXT -1 5 " and " }{TEXT 263 1 "a" }{TEXT -1 147 " are not specified \+ as arguments to kfreerec, then the resulting recurrence equation conta ins these variable names as local variables. All symbolic " }{XPPEDIT 18 0 "a[i,j]" "6#&%\"aG6$%\"iG%\"jG" }{TEXT -1 62 " in the resulting r ecurrence equation are arbitrary constants." }}{PARA 15 "" 0 "" {TEXT -1 74 "The procedure fasenmyer delivers an upward recurrence equation \+ ,ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormaltextwarnerrorlistitembulletfunctextendgospextensalgorithmindefinitummatusagcallsequencparameterexpressionnamesummatvariablexpressrepresntingupperlowerboundintegsynopsidescriptimplementatcalculathypergeometricupwardantidifferencjgfoldfgkgwhenevsuchantidifferenceexistcaseprocedurcanuseddefinitesumssumsumgmgngdoesdependoccurrhypergeometricwithrespectjgfrationaltypicalratioproductationalfunctionpowerfactorialbinomialcoefficientpochhammsymbollineartheirargumentimplementatsupporttypeggsecondarguminvokedreturnhypergeometritermrespectapplicablformdefinitdeterminapplinvokthirdtakennifresultnoteroccurprovmessageshowionhypergeometricexamplreadhsummplvpackagycopyrightwolframkoepfuniversitkasselgsiamreviewproblemkgffactorialgngfbinomialgpochhammbgpochhammergxtendseealsoalsosumrecursVibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctclosedformsumrecurszeilbergalgorithmusagcallsequencparameterexpressnamesummatvariablrecurrencsynopsidescriptthesfunctionimplementatwhilecalculatupwardequatiosumsgngsumgfgkgagbghypergeometricwithrespectclosdformtrydeterminhypergeometrictermequalhypergeometrickgfrationalwittypicalcasratioproductpowerfactorialbinomialcoefficientpochhammsymbolinteglineartheirargumentsupporttypewithoutanyboundalwayhomogeneousassumfinitnaturalzeroutsidranginfinitsummationshouldusenthuenforcingcomputatinhomogeneoupartequaalsohsumtheorallprocedurespossiblobtaincertificatallowsimplposterioproofequatusingoptionalargumcertificattruewillreturnentrfirstcomputsecondcertificatnotelookonlyionfirstorderwhilsearchupmaxordlobalsetdefaultbutmaychangpositivevaluyoucanoptionalrecordnonnegatik -1 3 "],[" }{XPPEDIT 18 0 "L[1]" "6#&%\"LG6#\"\"\"" }{TEXT -1 5 ",...," }{XPPEDIT 18 0 "L[q]" "6#&%\"LG6#%\"qG" }{TEXT -1 2 "]," }{XPPEDIT 18 0 "z" "6#%\"zG" } {TEXT -1 1 "," }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 2 ")=" } {XPPEDIT 18 0 "product(pochhammer(U[i],k),i=1..p)/product(pochhammer(L [i],k),i=1..q)" "6#*&-%(productG6$-%+pochhammerG6$&%\"UG6#%\"iG%\"kG/F -;\"\"\"%\"pGF1-F%6$-F(6$&%\"LG6#F-F./F-;F1%\"qG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "z^k/k!" "6#*&)%\"zG%\"kG\"\"\"-%*factorialG6#F&!\"\" " }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Ma ple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfr am~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "hyperterm([a,b],[c],z,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* ,-%+pochhammerG6$%\"aG%\"kG\"\"\"-F%6$%\"bGF(F)-F%6$%\"cGF(!\"\")%\"zG F(F)-%*factorg Sequence:" }{TEXT -1 52 "\n hyperterm(U, L, z, \+ k);\n Hyperterm(U, L, z, k);" }}{PARA 0 "" 0 "" {TEXT 26 11 "Paramet ers:" }{TEXT -1 4 "\n " }{TEXT 23 6 "U,L - " }{TEXT -1 43 "lists of \+ the upper and lower parameters\n " }{TEXT 23 6 "z - " }{TEXT -1 20 "evaluation point\n " }{TEXT 23 6 "k - " }{TEXT -1 20 "name of \+ the variable" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Descript ion:" }}{PARA 15 "" 0 "" {TEXT -1 244 "This function is a shorthand fo r a hypergeometric term of variable k where U and L denote the lists o f upper and lower parameters, and z is the evaluation point. The proce dure Hyperterm is the corresponding inert form which remains unevaluat ed." }}{PARA 15 "" 0 "" {TEXT -1 63 "hyperterm(U,L,z,k) is the summand of the hypergeometric series " }{HYPERLNK 17 "hypergeom(U,L,z,k)" 2 " hypergeom" "" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 11 "hyperter m([" }{XPPEDIT 18 0 "U[1]" "6#&%\"UG6#\"\"\"" }{TEXT -1 4 ",..." } {XPPEDIT 18 0 "U[p]" "6#&%\"UG6#%\"pG" }{TEXT );\n kfreerec(f, k, n, kmax, nmax, F);\n kfreerec(f, k, \+ n, kmax, nmax, F, a);\n fasenmyer(f, k, s(n), nmax);" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 4 "f - " }{TEXT -1 17 "an expression\n " }{TEXT 23 4 "k - " }{TEXT -1 34 "a n ame, the summation variable\n " }{TEXT 23 4 "n - " }{TEXT -1 35 "a n ame, the recurrence variable\n " }{TEXT 23 7 "F, s - " }{TEXT -1 35 "a name, the recurrence function\n " }{TEXT 256 10 "kmax, nmax" } {TEXT -1 62 " - nonnegative integers, the recurrence order wrt. k and \+ n\n " }{TEXT 265 1 "a" }{TEXT -1 30 " - a name, arbitrary constants " }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }} {PARA 15 "" 0 "" {TEXT -1 61 "The functions are an implementation of F asenmyer's algorithm." }}{PARA 15 "" 0 "" {TEXT -1 36 "The procedure k freerec calculates a " }{TEXT 260 1 "k" }{TEXT -1 29 "-free recurrence equation in " }{TEXT 258 1 "k" }{TEXT -1 5 " and " }{TEXT 259 1 "n" } {TEXT -1 29 " for the hypergeometric termCSTYLE "" -1 264 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 265 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 43 "kfreerec, fasenmyer - Fasenmyer's algorithm" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 143 "\n kfreerec(f, k, n, \+ kmax, nmax#Q0{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 " " 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }c1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 5 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "HolonomicDE (SimpleDE)" }{TEXT 30 38 " - find a linear differential equation" }}{PARA 256 "" 0 "usage " {TEXT -1 70 "Calling Sequences (SimpleDE is an alias for HolonomicDE and obsolete):" }}{PARA 0 "" 0 "" {TEXT -1 16 " HolonomicDE" } {TEXT 256 9 "(f,F(x))\n" }{TEXT -1 16 " HolonomicDE" }{TEXT 257 10 "(f,F(x),n)" }}{PARA 0 "" 0 "" {TEXT -1 17 " HolonomicDE(" } {TEXT 25 3 "f,x" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 17317 "FPS[SimpleR E]" 2 "FPS[SimpleRE]" "" }{TEXT 297 2 ", " }{HYPERLNK 17 "FPS[standard sum]" 2 "standardsum" "" }{TEXT 305 2 ", " }{HYPERLNK 17 "inifcns" 2 " inifcns" "" }{TEXT 298 2 ", " }{HYPERLNK 17 "taylor" 2 "taylor" "" } {TEXT 299 2 ", " }{HYPERLNK 17 "series" 2 "series" "" }{TEXT 300 2 ", \+ " }{HYPERLNK 17 "Sum" 2 "Sum" "" }}}}{MARK "12 1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } > " 0 "" {MPLTEXT 1 0 93 "FPS((-1/2*x+1/6*x^3)*arcta n(x)+(-1/4*x^2+1/12)*\nln(x^2+1)+5/12*x^2+1/4, x=0, hypergeometric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"\"%F%*&F$F%-%$SumG6$*.)! \"\"%\"kGF%,&F.F%\"\"#F%F-,&*&F0F%F.F%F%F%F%F-,&F.F%F%F%F-,&\"\"$F%*&F 0F%F.F%F%F-)%\"xG,&*&F0F%F.F%F%F&F%F%/F.;\"\"!%)infinityGF%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 297 "" 0 "seealso" {TEXT 294 10 "See Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "FP S" 2 "FPS" "seealso" }{TEXT 295 2 ", " }{HYPERLNK 17 "FPS[HolonomicDE] " 2 "FPS[HolonomicDE]" "" }{TEXT 296 2 ", " }{HYPERLNK (ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormaltextwarningerrorlistitembulletfunctwzcertificatwilfzeilbergmethodusagcallsequencparameterexpressnamesummatvariablrecurrencsynopsidescriptimplementatzeilbrgerwzcanusedprovidentitformsgsuminfinitsumgfgkginfinitygassumhypergeometrictermwithfinitesupportwrtidenticalzerooutsidfinitintervaleachsuccessfulreturnrationalrgfulfilfollowequatkgfsummoverallintegerusinsomefurthknowledgaboutpolewegetsimplvalidfailmessagthusmayidentitiinfinitycghypergometriccontainsumsbounddependhavedoapplprocedurzcertificatcheckinfinitygfexamplreadhsummplvpackagmapleycopyrightwolframkoepfuniversitkasselgletbinomialbinomialgxgfngfapplicatdixongammabinomialgbgfcgfgammagactorialgfcyieldwzcertificateherecannotprovedinomialextendmethodseealsoalsogospergospkfreerecfasenmyerfasenmyclosedformrecurssumrecurs{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 " " 2age~\"Hypergeometric~Summation\",~Mapl e~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-200 2,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Calculate a recurrence for " }{XPPEDIT 18 0 "F(n,k)=k*bin omial(n,k)" "6#/-%\"FG6$%\"nG%\"kG*&F(\"\"\"-%)binomialG6$F'F(F*" } {TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "kfreerec(k*binomia l(n,k),k,n,1,1,F,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(&%\"aG6$ \"\"\"F)F)%\"nGF)-%\"FG6$,&F*F)F)F),&%\"kGF)F)F)F)F)*(F&F)F.F)-F,6$F*F /F)!\"\"*(F&F)F.F)-F,6$F*F0F)F4\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(a[1,1]=1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /,(*&-%\"FG6$,&%\"nG\"\"\"F+F+,&%\"kGF+F+F+F+F*F+F+*&F)F+-F'6$F*F,F+! \"\"*&F)F+-F'6$F*F-F+F1\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "C alculate a recurrence for the sum " }{XPPEDIT 18 0 "s(n)=sum(k*binomia l(n,k),k=-infinity..infinity)" "6#/-%\"sG6#%\"nG-%$sumG6$*&%\"kG\"\"\" -%)binomialG6$F'F,F-/F,;,$%)infinityG!\"\"F4" }{TEXT -1 1 ":" }}{PARA for the sum" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "s(n)=sum(f(n,k),k=-inf inity..infinity)" "6#/-%\"sG6#%\"nG-%$sumG6$-%\"fG6$F'%\"kG/F.;,$%)inf inityG!\"\"F2" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 126 "by cal culating (via kfreerec) a recurrence equation for the summand and then summing the resulting equation over all integers " }{XPPEDIT 18 0 "k " "6#%\"kG" }{TEXT -1 55 ". (Note that the procedure assumes that the \+ input term " }{TEXT 264 1 "f" }{TEXT -1 116 " has finite support.) The output is a homogeneous recurrence equation, and is a function of the recurrence variable " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "s(n)" "6#-%\"sG6#%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "s(n+1)" "6#-%\"sG6#,&%\"nG\"\"\"F(F(" }{TEXT -1 6 ",..., " } {XPPEDIT 18 0 "s(n+nmax)" "6#-%\"sG6#,&%\"nG\"\"\"%%nmaxGF(" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPack+collectcomcombcombin  combinatocomm4commandcomment  communicat compcompar completcomplete complex  complexit compucomput?computat# conconclud  conferenc consid oibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalerrorbulletitemnormalfunctrecporechyppetkovsekalgorithmusagcallsequencrecpolyrehyperparameterequatalgebraicexpressrecurrencnamevariablsynopsidescriptthesfunctionimplementatprocedursearchpolynomialhypergeometrictersolutiongivenhomogeneouwithcoefficientngorrespondspecialversionthatcanonlyusedequationordersetcontainingallpossiblnosolutexistemptreturnnotefgrecurrencdobutrathrationalratioanydeltadeltagdeltagtheyconsiderarbitrarconstantthoslocalexamplreadhsummplvpackagsummatycopyrightwolframkoepfunivsitykasselgherewzmethodfailwzcertificatsumrecursdgminimalsubinomialsumgkgfbinomialgngfdgfllpositintegersumrecursregsgsumrecursipffcflfkffnfseealsoalsokfreerecfasenmyclosedformTEXT 259 19 "hypergeometric term" }{TEXT -1 17 " with respect to " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 3 " if" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(k+1)/f(k)" "6#*&- %\"fG6#,&%\"kG\"\"\"F)F)F)-F%6#F(!\"\"" }}{PARA 14 "" 0 "" {TEXT -1 28 "is rational with respect to " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 232 ". This is typically the case for ratios of products of rationa l functions, powers, factorials, binomial coefficients, and Pochhammer symbols that are integer-linear in their arguments. The implementatio n supports this type of input." }}{PARA 15 "" 0 "" {TEXT -1 35 "If gos per returns the error message" }}{PARA 8 "" 0 "" {TEXT -1 43 "Error, ( in gosper) algorithm not applicable" }}{PARA 14 "" 0 "" {TEXT -1 36 "y ou might want to try the procedure " }{HYPERLNK 17 "extended_gosper" 2 "extended_gosper" "" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 29 "If gosper returns the message" }}{PARA 8 "" 0 "" {TEXT -1 63 "Error, \+ (in gosper) no hypergeometric term antidifference exi " } {TEXT 23 6 "f - " }{TEXT -1 17 "an expression\n " }{TEXT 23 6 "k \+ - " }{TEXT -1 30 "a name, the summation variable" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 92 "This function is an implementation of Gosper's algorithm for in definite summation (see also " }{HYPERLNK 17 "hsum[theory]" 2 "hsum[th eory]" "" }{TEXT -1 20 "), and calculates a " }{TEXT 257 14 "hypergeom etric" }{TEXT -1 1 " " }{TEXT 256 8 "(upward)" }{TEXT -1 1 " " }{TEXT 258 14 "antidifference" }{TEXT -1 26 " of a hypergeometric term " } {XPPEDIT 18 0 "f(k)" "6#-%\"fG6#%\"kG" }{TEXT -1 90 " whenever such an antidifference exists. I.e. the procedure returns a hypergeometric te rm " }{XPPEDIT 18 0 "g(k)" "6#-%\"gG6#%\"kG" }{TEXT -1 5 " with" }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "f(k)=g(k+1)-g(k)" "6#/-%\"fG6#%\"kG,& -%\"gG6#,&F'\"\"\"F-F-F--F*6#F'!\"\"" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 14 "An expression " }{XPPEDIT 18 0 "f(k)" "6#-%\"fG6#%\"kG " }{TEXT -1 6 " is a " }{00 1 0 0 255 1 0 0 0 0 0 0 1 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "B ullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 52 "gosper - Gosper's algorithm for indefinite summation" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 17 "\n gosper(f, k );" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n1ngengfX nheinrichnhomo nhypertermnifnit nitenity nitygnk#nlnln nmax nmaxg nmaxgf 6 "" 0 "" {XPPEDIT 18 0 "product(qpochhammer(a*q^i,q^k,n),i=0..k-1) = qpochhammer(a,q,k*n) " "6#/-%(productG6$-%,qpochhammerG6%*&%\"aG\"\"\")%\"qG%\"iGF,)F.%\"kG %\"nG/F/;\"\"!,&F1F,F,!\"\"-F(6%F+F.*&F1F,F2F," }{TEXT 263 1 " " }}} {SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boeing~&~Wolf ram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "f:= qpochhammer(a*q^(-k*n),q,n)-qpochhammer(q/a,q,k* n)/\nqpochhammer(q/a,q,k*n-n)*(-a)^n*q^(binomial(n,2)-k*n^2);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"fG,&-%,qpochhammerG6%*&%\"aG\"\"\" )%\"qG,$*&%\"kGF+%\"nGF+!\"\"F+F-F1F+**-F'6%*&F-F+F*F2F-F/F+-F'6%F6F-, &F/F+F1F2F2),$F*F2F1F+)F-,&-%)binomialG6$F1\"\"#F+*&F0F+)F1FAF+F2F+F2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qsimplify(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 \evenever exexactexamexamp exampltexcept exist3exp/pward) antidifference of " }{XPPEDIT 18 0 "f(k)" "6#-%\"fG6#% \"kG" }{TEXT -1 7 " (i.e. " }{TEXT 258 1 "f" }{TEXT -1 18 ") with resp ect to " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 17 ", if applicable. \+ " }}{PARA 15 "" 0 "" {TEXT -1 36 "If the second argument has the form \+ " }{XPPEDIT 18 0 "k = m..n" "6#/%\"kG;%\"mG%\"nG" }{TEXT -1 22 " then \+ the definite sum" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "sum(f(k),k=m..n) " "6#-%$sumG6$-%\"fG6#%\"kG/F);%\"mG%\"nG" }}{PARA 14 "" 0 "" {TEXT -1 45 "is determined if Gosper's algorithm applies. " }}{PARA 15 "" 0 "" {TEXT -1 99 "If extended_gosper is invoked with three arguments the n the third argument is taken as the integer " }{XPPEDIT 18 0 "j" "6#% \"jG" }{TEXT -1 8 ", and a " }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 31 "-fold upward antidifference of " }{XPPEDIT 18 0 "f(k)" "6#-%\"fG6# %\"kG" }{TEXT -1 30 " is returned whenever it is a " }{XPPEDIT 18 0 "j " "6#%\"jG" }{TEXT -1 40 "-fold hypergeometric term.\nIf the result" } }{PARA 8 "" 0 "" {TEXT -1 63 "Error]l coefficients, and Poch hammer symbols that are rational-linear in their arguments. The implem entation supports this type of input. " }}{PARA 15 "" 0 "" {TEXT -1 14 "An expression " }{XPPEDIT 18 0 "g(k)" "6#-%\"gG6#%\"kG" }{TEXT -1 36 " is called upward antidifference of " }{XPPEDIT 18 0 "f(k)" "6#-% \"fG6#%\"kG" }{TEXT -1 3 " if" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "f(k) =g(k+1)-g(k)" "6#/-%\"fG6#%\"kG,&-%\"gG6#,&F'\"\"\"F-F-F--F*6#F'!\"\" " }{TEXT 256 1 "." }}{PARA 15 "" 0 "" {TEXT -1 14 "An expression " } {XPPEDIT 18 0 "g(k)" "6#-%\"gG6#%\"kG" }{TEXT -1 11 " is called " } {XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 35 "-fold upward antidifference \+ of f if" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "f(k)=g(k+j)-g(k)" "6#/-%\" fG6#%\"kG,&-%\"gG6#,&F'\"\"\"%\"jGF-F--F*6#F'!\"\"" }{TEXT 257 1 "." } }{PARA 15 "" 0 "" {TEXT -1 23 "If the second argument " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 143 " is a name, and extended_gosper is invo ked with two arguments, then extended_gosper returns the hypergeometri c term (u7G6#%\"kG" }{TEXT -1 107 " whenever such an antidifferen ce exists. In this case, the procedure can be used to calculate defini te sums" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sum(f(k),k =m..n)" "6#-%$sumG6$-%\"fG6#%\"kG/F);%\"mG%\"nG" }{TEXT -1 1 " " }} {PARA 14 "" 0 "" {TEXT -1 9 "whenever " }{XPPEDIT 18 0 "f(k)" "6#-%\"f G6#%\"kG" }{TEXT -1 43 " does not depend on variables occurring in " } {XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n" "6# %\"nG" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 28 "An expression \+ f is called a " }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 48 "-fold hyper geometric expression with respect to " }{XPPEDIT 18 0 "k" "6#%\"kG" } {TEXT -1 4 " if " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "f(k+j)/f(k)" "6#* &-%\"fG6#,&%\"kG\"\"\"%\"jGF)F)-F%6#F(!\"\"" }}{PARA 14 "" 0 "" {TEXT -1 28 "is rational with respect to " }{XPPEDIT 18 0 "k" "6#%\"kG" } {TEXT -1 234 ". This is typically the case for ratios of products of r ational functions, powers, factorials, binomia8 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 77 "extended_gosper - an extension of Gosper's algorithm for indefinite s ummation" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" } {TEXT -1 86 "\n extended_gosper(f, k);\n extended_gosper(f, k=m..n );\n extended_gosper(f, k, j);" }}{PARA 0 "" 0 "" {TEXT 26 11 "Param eters:" }{TEXT -1 4 "\n " }{TEXT 23 6 "f - " }{TEXT -1 17 "an expr ession\n " }{TEXT 23 6 "k - " }{TEXT -1 34 "a name, the summation \+ variable\n " }{TEXT 23 6 "m,n - " }{TEXT -1 62 "expressions, represe nting upper and lower summation bounds\n " }{TEXT 23 6 "j - " } {TEXT -1 10 "an integer" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 135 "This function is an implementation of an extension of Gosper's algorithm, and calculates \+ a hypergeometric (upward) antidifference of a " }{XPPEDIT 18 0 "j" "6# %\"jG" }{TEXT -1 32 "-fold hypergeometric expression " }{XPPEDIT 18 0 "f(k)" "6#-%\"f91 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item " 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 259 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0:ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctsumtohypexpresinfinitsumhypergeometricusagcallsequencparameterexpressnamesummatvariablsynopsidescriptconvertsumgfgkginfinityginvolvexponentialfactorialbinomialcoefficientpochhammsymbolintonotatioreturninerthypergeomalsoprocedurusessomeassumptgivenindeterminatsummandspecifiviaassumexamplreadhsummplvpackagycopyrightwolframkoepfunivsitykasselgerbinomialhypergeomgngtohyphypergeomgngfhereusconsidbinomialgfirstverswithoutanyresultepresentatwithwrongboundlefthandsincilateralbinomialgintegnowprogramcanshiftntegwegetcorrectrepresentatirgfseealsohypertermhypertermoibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalerrorbulletitemnormalfunctrecporechyppetkovsekalgorithmusagcallsequencrecpolyrehyperparameterequatalgebraicexpressrecurrencnamevariablsynopsidescriptthesfunctionimplementatprocedursearchpolynomialhypergeometrictersolutiongivenhomogeneouwithcoefficientngorrespondspecialversionthatcanonlyusedequationordersetcontainingallpossiblnosolutexistemptreturnnotefgrecurrencdobutrathrationalratioanydeltadeltagdeltagtheyconsiderarbitrarconstantthoslocalexamplreadhsummplvpackagsummatycopyrightwolframkoepfunivsitykasselgherewzmethodfailwzcertificatsumrecursdgminimalsubinomialsumgkgfbinomialgngfdgfllpositintegersumrecursregsgsumrecursipffcflfkffnfseealsoalsokfreerecfasenmyclosedform(ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalasicqgfunctioncallsequencqpochhamminfinitnqpochhammnqfacnqbracketnqfactorialnqbinomialdescriptqpochhammerfuncttakeargumentfirstparametanyalgebraicexpresssecondtypenameraisotintegerpowerthirdvariablalgebraicrepresingintegvalusymbolicdefinproductqpochhammergaginfinitygproductgjgfergkgpiecewisegproductgqfacunctsynonymqbracketbothcanvaliddefinitqbracketsgqbinomialusesexpressionsrepresentlastqbinomialgngqfactorialqfactorialqfactorialgqgammatwzgexamplreadqsummplppackaghypergeometricsummatcocopyrightharaldboeingwolframkoepuniversitkasselgrocedurqsimpcombprovidsimplificatmechanismallthosunctionagfqgfalsoqhypertermqpsihyperterm {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bul let Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } 0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 260 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 ""  9 "Examples:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Mapl e~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-200 2,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Binomial Identity:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " zeilberger(binomial(n,k),k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,&*&\"\"#\"\"\"-%\"sG6#%\"nGF'F'-F)6#,&F+F'F'F'!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Dixon's identity," }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sum((-1)^k*binomial(n+b,n+k)*binomial(n+c,c+k)*binomial (b+c,b+k),k=-infinity..infinity)=GAMMA(b+c+n+1)/(n!*GAMMA(b+1)*GAMMA(c +1))" "6#/-%$sumG6$**),$\"\"\"!\"\"%\"kGF*-%)binomialG6$,&%\"nGF*%\"bG F*,&F1F*F,F*F*-F.6$,&F1F*%\"cGF*,&F7F*F,F*F*-F.6$,&F2F*F7F*,&F2F*F,F*F */F,;,$%)infinityGF+F@*&-%&GAMMAG6#,*F2F*F7F*F1F*F*F*F**(-%*factorialG 6#F1F*-FC6#,&F2F*F*F*F*-FC6#,&F7F*F*F*F*F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT - while sumrecursion searches for recurrence equation s of order one up to " }{TEXT 256 8 "MAXORDER" }{TEXT -1 312 ". This g lobal variable is set by default to 5 but may be changed to any positi ve integer value or infinity. For sumrecursion you can also use the op tional argument recorder=r..s or recorder =r with nonnegative integers r and to look for a recurrence equation in the range r..s or of order exactly r respectively." }}{PARA 15 "" 0 "" {TEXT -1 193 "The procedu res closedform and Closedform try to determine a recurrence equation o f first order via the procedure zeilberger. If successful this recurre nce equation is solved with initial value " }{XPPEDIT 18 0 "s(0)" "6#- %\"sG6#\"\"!" }{TEXT -1 100 " and the solution - a hypergeometric term - is returned: The procedure closedform uses the function " } {HYPERLNK 17 "hyperterm" 2 "hyperterm" "" }{TEXT -1 91 " to return the solution whereas the procedure Closedform uses the inert function Hyp erterm." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26A the summa tion bounds are " }{TEXT 258 11 "not natural" }{TEXT -1 162 ", one sho uld always use the call\n sumrecursion(f,k=a..b,s(n));\nthus enfo rcing the computation of the inhomogeneous part of the recurrence equa tion (see also " }{HYPERLNK 17 "hsum[theory]" 2 "hsum[theory]" "" } {TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXT -1 60 "With all these procedu res it is also possible to obtain the " }{TEXT 261 20 "rational certif icate" }{TEXT -1 6 " (see " }{HYPERLNK 17 "hsum[theory]" 2 "hsum[theor y]" "" }{TEXT -1 239 ") which allows a simple a posterio proof of the \+ calculated recurrence equation. When using the optional argument certi ficate=true, sumrecursion will return a list with two entries. The fir st is the computed recurrence equation for the sum " }{XPPEDIT 18 0 "s (n)" "6#-%\"sG6#%\"nG" }{TEXT -1 29 " and the second entry is the " } {TEXT 262 20 "rational certificate" }{TEXT -1 1 "." }}{PARA 15 "" 0 " " {TEXT -1 145 "Note that zeilberger looks only for a recurrence equat ion of first orderB"n" "6#%\"nG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 232 ". This is typically the cas e for ratios of products of rational functions, powers, factorials, bi nomial coefficients, and Pochhammer symbols that are integer-linear in their arguments. The implementation supports this type of input." }} {PARA 15 "" 0 "" {TEXT -1 84 "The output of sumrecursion (without any \+ summation bounds) or zeilberger is always a " }{TEXT 257 22 "homogeneo us recurrence" }{TEXT -1 26 ". I.e. it is assumed that " }{XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 49 " has finite support an d the summation bounds are " }{TEXT 259 7 "natural" }{TEXT -1 2 " (" } {XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 38 " is zero o utside the summation range):" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "Sum(f (n,k),k=a[n]..b[n])=Sum(f(n,k),k=-infinity..infinity)" "6#/-%$SumG6$-% \"fG6$%\"nG%\"kG/F+;&%\"aG6#F*&%\"bG6#F*-F%6$-F(6$F*F+/F+;,$%)infinity G!\"\"F;" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 28 "IfC bG6#F'" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 35 " is hyperg eometric with respect to " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 98 ", the functions close dform and Closedform try to determine a hypergeometric term that is eq ual to " }{XPPEDIT 18 0 "s(n)" "6#-%\"sG6#%\"nG" }{TEXT -1 1 "." }} {PARA 15 "" 0 "" {TEXT -1 14 "An expression " }{XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 11 " is called " }{TEXT 260 14 "hyper geometric" }{TEXT -1 23 " term with respect to (" }{XPPEDIT 18 0 "n" " 6#%\"nG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 4 ") i f" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "f(n+1,k)/f(n,k)" "6#*&-%\"fG6$,& %\"nG\"\"\"F)F)%\"kGF)-F%6$F(F*!\"\"" }{TEXT -1 22 " and \+ " }{XPPEDIT 18 0 "f(n,k+1)/f(n,k)" "6#*&-%\"fG6$%\"nG,&%\"kG\"\"\" F*F*F*-F%6$F'F)!\"\"" }}{PARA 14 "" 0 "" {TEXT -1 29 "are rational wit h respect to " }{XPPEDIT 18 0 D0 "" {TEXT 26 10 "Function: " }{TEXT -1 61 "closedform, sumrecursion, zeilberger - Zeilberger's algorithm" }} {PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 128 "\n closedform(f,k,n);\n Closedform(f,k,n);\n sumrecursion(f,k,s(n) );\n sumrecursion(f,k=a..b,s(n));\n zeilberger(f,k,s(n));" }} {PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 4 "f - " }{TEXT -1 17 "an expression\n " }{TEXT 23 4 "k - " } {TEXT -1 34 "a name, the summation variable\n " }{TEXT 23 4 "n - " } {TEXT -1 35 "a name, the recurrence variable\n " }{TEXT 23 4 "s - " }{TEXT -1 31 "a name, the recurrence function" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 148 "These functions are implementations of Zeilberger's algorithm. Wh ile sumrecursion and zeilberger calculate an upward recurrence equatio n for the sum" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "s(n)=sum(f(n,k),k=a[ n]..b[n])" "6#/-%\"sG6#%\"nG-%$sumG6$-%\"fG6$F'%\"kG/F.;&%\"aG6#F'&%\"E 1 "" {XPPMATH 20 "6#>%$re cG,&*(%\"nG\"\"\"%\"kGF(-%)binomialG6$,&F'F(F(F(F)F(!\"\"**\"\"#F(F-F( F)F(-F+6$F'F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG,$*,,&%\"nG \"\"\"F)F)F),&%\"kGF)F)!\"\"F),(F(F,F)F,F+F)F,F+F)-%)binomialG6$F(F+F) F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simpcomb(rec-(subs(k= k+1,G)-G));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "gosper" 2 "gosper" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "hyperterm" 2 "hyperterm" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intrecursion" 2 "intrecursion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "kfreerec and fasenmyer" 2 "fasenmyer" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "sumdeltanabla" 2 "sumdeltanabla" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffeq" 2 "sumdiffeq" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "sumdiffrule and sumintrule" 2 "sumdiffrule" "" }{TEXT -1 2 ", " }{HYPERLNK 17b*FGF*F/F*F*F)F*F*F:F*,&*&F+F*FGF*F/F)F*F*,(F)F*F*F*FGF/!\" #,(F)F*F+F*FGF/FM" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Calculate a \+ recurrence equation for " }{XPPEDIT 18 0 "s(n)=sum(k*binomial(n,k),k=0 ..n-2)" "6#/-%\"sG6#%\"nG-%$sumG6$*&%\"kG\"\"\"-%)binomialG6$F'F,F-/F, ;\"\"!,&F'F-\"\"#!\"\"" }{TEXT -1 1 ":" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "tmp:=sumrecursion(k*binomial(n,k),k=0..n-2,s(n), certificate=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tmpG7$/,&*&% \"nG\"\"\"-%\"sG6#,&F)F*F*F*F*!\"\"*(\"\"#F*F.F*-F,6#F)F*F*,$*(,&F)F*F *F/F*F)F*F.F*F/,$*(F.F*,&%\"kGF*F*F/F*,(F)F/F*F/F:F*F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Now prove the correspoding recurrence equ ation for the summand " }{XPPEDIT 18 0 "k*binomial(n,k)" "6#*&%\"kG\" \"\"-%)binomialG6$%\"nGF$F%" }{TEXT -1 6 " (see " }{HYPERLNK 17 "hsum[ theory]" 2 "hsum[theory]" "" }{TEXT -1 2 "):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "rec:= eval(subs(s=(n->k*binomial(n,k)),lhs(tmp[1]))); \nG:= tmp[2]*k*binomial(n,k);" }}{PARA 11 ""H&6%7%*&F.F-F)F-*&F. F-F*F-,&F)F-F*F-7$,(F)F-F*F-*&F-F-F.!\"\"F-,&*&F.F-F)F-F-*&F.F-F*F-F-F /" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Closedform(hyperterm([a,b],[a+ b+1/2],x,j)*\n hyperterm([a,b],[a+b+1/2],x,k-j),j,k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*HypertermG6&7%,$*&\"\"#\"\"\"%\"bGF *F*,$*&F)F*%\"aGF*F*,&F.F*F+F*7$,&*&F)F*F.F*F**&F)F*F+F*F*,(F.F*F+F*#F *F)F*%\"xG%\"kG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sumrecur sion(binomial(n,k)^3,k,f(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*& ),&%\"nG\"\"\"\"\"#F)F*F)-%\"fG6#F'F)!\"\"*&,(*&\"\"(F))F(F*F)F)*&\"#@ F)F(F)F)\"#;F)F)-F,6#,&F(F)F)F)F)F)*(\"\")F))F9F*F)-F,6#F(F)F)\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "sumrecursion(binomial(n,k) ^2*binomial(2*k,n),k,s(n),certificate=true,recorder=2);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7$/,(*&),&%\"nG\"\"\"\"\"#F*F+F*-%\"sG6#F(F*!\" \"*&,(*&\"\"(F*)F)F+F*F**&\"#@F*F)F*F*\"#;F*F*-F-6#,&F)F*F*F*F*F**(\" \")F*)F:F+F*-F-6#F)F*F*\"\"!*0,(*&\"\"$F*F)F*F*\"\"'F**&F+F*%\"kGF*F/F *FGF+,(*&F+FI1 34 "a recurrence equation for the sum " }{XPPEDIT 18 0 "s(n)" "6#-%\"sG6#%\"nG" }{TEXT -1 25 " on the left hand side..." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "zeilberger((-1)^k*binomial(n+b,n+k) *binomial(n+c,c+k)*binomial(b+c,b+k),k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,*%\"bG\"\"\"%\"nGF(F(F(%\"cGF(F(-%\"sG6#F)F(F(*&, &F(!\"\"F)F0F(-F,6#,&F)F(F(F(F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "...and the corresponding right hand side determined by cl osedform:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "closedform((-1)^k*bino mial(n+b,n+k)*binomial(n+c,c+k)*binomial(b+c,b+k),k,n);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*(-%)binomialG6$,&%\"bG\"\"\"%\"cGF)F(F)-%+pochh ammerG6$,(F(F)F*F)F)F)%\"nGF)-%*factorialG6#F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "An application of Closedform to deduce Clausen' s formula:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "Hypergeom([a,b],[a+b+1, 2],x)^2=Hypergeom([2*a,2*b,a+b],[a+b+1/2,2*a+2*b],x)" "6#/*$-%*Hyperge omG6%7$%\"aG%\"bG7$,(F)\"\"\"F*F-F-F-\"\"#%\"xGF.-FJ 9 "Examples:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Mapl e~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-200 2,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Binomial Identity:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " zeilberger(binomial(n,k),k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,&*&\"\"#\"\"\"-%\"sG6#%\"nGF'F'-F)6#,&F+F'F'F'!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Dixon's identity," }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sum((-1)^k*binomial(n+b,n+k)*binomial(n+c,c+k)*binomial (b+c,b+k),k=-infinity..infinity)=GAMMA(b+c+n+1)/(n!*GAMMA(b+1)*GAMMA(c +1))" "6#/-%$sumG6$**),$\"\"\"!\"\"%\"kGF*-%)binomialG6$,&%\"nGF*%\"bG F*,&F1F*F,F*F*-F.6$,&F1F*%\"cGF*,&F7F*F,F*F*-F.6$,&F2F*F7F*,&F2F*F,F*F */F,;,$%)infinityGF+F@*&-%&GAMMAG6#,*F2F*F7F*F1F*F*F*F**(-%*factorialG 6#F1F*-FC6#,&F2F*F*F*F*-FC6#,&F7F*F*F*F*F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -K while sumrecursion searches for recurrence equation s of order one up to " }{TEXT 256 8 "MAXORDER" }{TEXT -1 312 ". This g lobal variable is set by default to 5 but may be changed to any positi ve integer value or infinity. For sumrecursion you can also use the op tional argument recorder=r..s or recorder =r with nonnegative integers r and to look for a recurrence equation in the range r..s or of order exactly r respectively." }}{PARA 15 "" 0 "" {TEXT -1 193 "The procedu res closedform and Closedform try to determine a recurrence equation o f first order via the procedure zeilberger. If successful this recurre nce equation is solved with initial value " }{XPPEDIT 18 0 "s(0)" "6#- %\"sG6#\"\"!" }{TEXT -1 100 " and the solution - a hypergeometric term - is returned: The procedure closedform uses the function " } {HYPERLNK 17 "hyperterm" 2 "hyperterm" "" }{TEXT -1 91 " to return the solution whereas the procedure Closedform uses the inert function Hyp erterm." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26L/r{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 25 "Courier" 0 1 0 0 0 1 2 2 0 0 0 0 0 0 0 1 }{CSTYLE "Help Norma l" -1 30 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 " times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Head ing 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0XT -1 7 "Let " }{XPPEDIT 18 0 "s(n,x)=sum(f(n,k,x),x= -infinity..infinity)" "6#/-%\"sG6$%\"nG%\"xG-%$sumG6$-%\"fG6%F'%\"kGF( /F(;,$%)infinityG!\"\"F3" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 74 "The procedure sumdiffrule tries to determine a derivative rule of \+ the form" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(s(n,x),x)=sum(sigma[ j](n,x)*s(n+j,x),j=0..J)" "6#/-%%diffG6$-%\"sG6$%\"nG%\"xGF+-%$sumG6$* &-&%&sigmaG6#%\"jG6$F*F+\"\"\"-F(6$,&F*F6F4F6F+F6/F4;\"\"!%\"JG" } {TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 20 "The global variable " } {TEXT 256 8 "MAXORDER" }{TEXT -1 25 " gives the maximum shift " } {XPPEDIT 18 0 "s(n+j,x)" "6#-%\"sG6$,&%\"nG\"\"\"%\"jGF(%\"xG" }{TEXT -1 122 " that can appear on the right-hand side. By default it is set \+ to 5 but you can assign any positive integer or infinity to " }{TEXT 257 8 "MAXORDER" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 80 "The p rocedure sumintrule tries to determine an integration rule for the int egral" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "int(1 34 "a recurrence equation for the sum " }{XPPEDIT 18 0 "s(n)" "6#-%\"sG6#%\"nG" }{TEXT -1 25 " on the left hand side..." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "zeilberger((-1)^k*binomial(n+b,n+k) *binomial(n+c,c+k)*binomial(b+c,b+k),k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,*%\"bG\"\"\"%\"nGF(F(F(%\"cGF(F(-%\"sG6#F)F(F(*&, &F(!\"\"F)F0F(-F,6#,&F)F(F(F(F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "...and the corresponding right hand side determined by cl osedform:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "closedform((-1)^k*bino mial(n+b,n+k)*binomial(n+c,c+k)*binomial(b+c,b+k),k,n);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*(-%)binomialG6$,&%\"bG\"\"\"%\"cGF)F(F)-%+pochh ammerG6$,(F(F)F*F)F)F)%\"nGF)-%*factorialG6#F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "An application of Closedform to deduce Clausen' s formula:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "Hypergeom([a,b],[a+b+1, 2],x)^2=Hypergeom([2*a,2*b,a+b],[a+b+1/2,2*a+2*b],x)" "6#/*$-%*Hyperge omG6%7$%\"aG%\"bG7$,(F)\"\"\"F*F-F-F-\"\"#%\"xGF.-Ff 9 "Examples:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Mapl e~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-200 2,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Binomial Identity:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " zeilberger(binomial(n,k),k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,&*&\"\"#\"\"\"-%\"sG6#%\"nGF'F'-F)6#,&F+F'F'F'!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Dixon's identity," }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sum((-1)^k*binomial(n+b,n+k)*binomial(n+c,c+k)*binomial (b+c,b+k),k=-infinity..infinity)=GAMMA(b+c+n+1)/(n!*GAMMA(b+1)*GAMMA(c +1))" "6#/-%$sumG6$**),$\"\"\"!\"\"%\"kGF*-%)binomialG6$,&%\"nGF*%\"bG F*,&F1F*F,F*F*-F.6$,&F1F*%\"cGF*,&F7F*F,F*F*-F.6$,&F2F*F7F*,&F2F*F,F*F */F,;,$%)infinityGF+F@*&-%&GAMMAG6#,*F2F*F7F*F1F*F*F*F**(-%*factorialG 6#F1F*-FC6#,&F2F*F*F*F*-FC6#,&F7F*F*F*F*F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -Q while sumrecursion searches for recurrence equation s of order one up to " }{TEXT 256 8 "MAXORDER" }{TEXT -1 312 ". This g lobal variable is set by default to 5 but may be changed to any positi ve integer value or infinity. For sumrecursion you can also use the op tional argument recorder=r..s or recorder =r with nonnegative integers r and to look for a recurrence equation in the range r..s or of order exactly r respectively." }}{PARA 15 "" 0 "" {TEXT -1 193 "The procedu res closedform and Closedform try to determine a recurrence equation o f first order via the procedure zeilberger. If successful this recurre nce equation is solved with initial value " }{XPPEDIT 18 0 "s(0)" "6#- %\"sG6#\"\"!" }{TEXT -1 100 " and the solution - a hypergeometric term - is returned: The procedure closedform uses the function " } {HYPERLNK 17 "hyperterm" 2 "hyperterm" "" }{TEXT -1 91 " to return the solution whereas the procedure Closedform uses the inert function Hyp erterm." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26R the summa tion bounds are " }{TEXT 258 11 "not natural" }{TEXT -1 162 ", one sho uld always use the call\n sumrecursion(f,k=a..b,s(n));\nthus enfo rcing the computation of the inhomogeneous part of the recurrence equa tion (see also " }{HYPERLNK 17 "hsum[theory]" 2 "hsum[theory]" "" } {TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXT -1 60 "With all these procedu res it is also possible to obtain the " }{TEXT 261 20 "rational certif icate" }{TEXT -1 6 " (see " }{HYPERLNK 17 "hsum[theory]" 2 "hsum[theor y]" "" }{TEXT -1 239 ") which allows a simple a posterio proof of the \+ calculated recurrence equation. When using the optional argument certi ficate=true, sumrecursion will return a list with two entries. The fir st is the computed recurrence equation for the sum " }{XPPEDIT 18 0 "s (n)" "6#-%\"sG6#%\"nG" }{TEXT -1 29 " and the second entry is the " } {TEXT 262 20 "rational certificate" }{TEXT -1 1 "." }}{PARA 15 "" 0 " " {TEXT -1 145 "Note that zeilberger looks only for a recurrence equat ion of first orderS"n" "6#%\"nG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 232 ". This is typically the cas e for ratios of products of rational functions, powers, factorials, bi nomial coefficients, and Pochhammer symbols that are integer-linear in their arguments. The implementation supports this type of input." }} {PARA 15 "" 0 "" {TEXT -1 84 "The output of sumrecursion (without any \+ summation bounds) or zeilberger is always a " }{TEXT 257 22 "homogeneo us recurrence" }{TEXT -1 26 ". I.e. it is assumed that " }{XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 49 " has finite support an d the summation bounds are " }{TEXT 259 7 "natural" }{TEXT -1 2 " (" } {XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 38 " is zero o utside the summation range):" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "Sum(f (n,k),k=a[n]..b[n])=Sum(f(n,k),k=-infinity..infinity)" "6#/-%$SumG6$-% \"fG6$%\"nG%\"kG/F+;&%\"aG6#F*&%\"bG6#F*-F%6$-F(6$F*F+/F+;,$%)infinity G!\"\"F;" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 28 "IfT bG6#F'" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 35 " is hyperg eometric with respect to " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 98 ", the functions close dform and Closedform try to determine a hypergeometric term that is eq ual to " }{XPPEDIT 18 0 "s(n)" "6#-%\"sG6#%\"nG" }{TEXT -1 1 "." }} {PARA 15 "" 0 "" {TEXT -1 14 "An expression " }{XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 11 " is called " }{TEXT 260 14 "hyper geometric" }{TEXT -1 23 " term with respect to (" }{XPPEDIT 18 0 "n" " 6#%\"nG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 4 ") i f" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "f(n+1,k)/f(n,k)" "6#*&-%\"fG6$,& %\"nG\"\"\"F)F)%\"kGF)-F%6$F(F*!\"\"" }{TEXT -1 22 " and \+ " }{XPPEDIT 18 0 "f(n,k+1)/f(n,k)" "6#*&-%\"fG6$%\"nG,&%\"kG\"\"\" F*F*F*-F%6$F'F)!\"\"" }}{PARA 14 "" 0 "" {TEXT -1 29 "are rational wit h respect to " }{XPPEDIT 18 0 U0 "" {TEXT 26 10 "Function: " }{TEXT -1 61 "closedform, sumrecursion, zeilberger - Zeilberger's algorithm" }} {PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 128 "\n closedform(f,k,n);\n Closedform(f,k,n);\n sumrecursion(f,k,s(n) );\n sumrecursion(f,k=a..b,s(n));\n zeilberger(f,k,s(n));" }} {PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 4 "f - " }{TEXT -1 17 "an expression\n " }{TEXT 23 4 "k - " } {TEXT -1 34 "a name, the summation variable\n " }{TEXT 23 4 "n - " } {TEXT -1 35 "a name, the recurrence variable\n " }{TEXT 23 4 "s - " }{TEXT -1 31 "a name, the recurrence function" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 148 "These functions are implementations of Zeilberger's algorithm. Wh ile sumrecursion and zeilberger calculate an upward recurrence equatio n for the sum" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "s(n)=sum(f(n,k),k=a[ n]..b[n])" "6#/-%\"sG6#%\"nG-%$sumG6$-%\"fG6$F'%\"kG/F.;&%\"aG6#F'&%\"VVibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctclosedformsumrecurszeilbergalgorithmusagcallsequencparameterexpressnamesummatvariablrecurrencsynopsidescriptthesfunctionimplementatwhilecalculatupwardequatiosumsgngsumgfgkgagbghypergeometricwithrespectclosdformtrydeterminhypergeometrictermequalhypergeometrickgfrationalwittypicalcasratioproductpowerfactorialbinomialcoefficientpochhammsymbolinteglineartheirargumentsupporttypewithoutanyboundalwayhomogeneousassumfinitnaturalzeroutsidranginfinitsummationshouldusenthuenforcingcomputatinhomogeneoupartequaalsohsumtheorallprocedurespossiblobtaincertificatallowsimplposterioproofequatusingoptionalargumcertificattruewillreturnentrfirstcomputsecondcertificatnotelookonlyionfirstorderwhilsearchupmaxordlobalsetdefaultbutmaychangpositivevaluyoucanoptionalrecordnonnegatimuibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctsumdeltanablasumdeltanablavariatzeilbergalgorithmusagcallsequencsumdeltanablparameterexpressnamesummatvariabldifferencsumsynopsisdescriptthesfunctionimplementatalgorithminfinitpgngxgsumgfgkginfinitygdeltanablasearchequatformsigmadeltataueltalambdasigmagdeltagnablagtauglambdagxgfdefinnecessarhypergeometricwrtkgfhaverationalsumdeltahomogeneoustatabovexamplreadhsummplvpackaghypergeometricmapleycopyrightwolframkoepfuniversitkasselghahnhahntermpochhammerpochhammbetanhypertermalphasumdeltanablabetagsgalphagfngfalphagfnablagfwilsonwilsontermnpochhammwilsontermcgfdgfbgfagffdfdffsftabgseealsoalsosumrecursmrecurshypertermintrecursintdiffeqsumdiffeqsumdiffrulsumintrulfollows the name of the procedure, another \+ `_` and finally the name of the option. E.g. the procedure " }{TEXT 19 13 "qsumrecursion" }{TEXT -1 16 " has the option " }{TEXT 19 8 "rec range" }{TEXT -1 44 ", thus the corresponding global variable is " } {TEXT 19 23 "_qsumrecursion_recrange" }{TEXT -1 67 ". (If the option i s used by more than one procedure it starts with " }{TEXT 19 6 "_qsum_ " }{TEXT -1 38 ", followed by the name of the option.)" }}{PARA 0 "" 0 "" {TEXT -1 123 "In most cases the values that can be assigned to th ose global variables are the same as the possible values for the optio n." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 36 "Description of global vari ables for:" }}{SECT 0 {PARA 4 "" 0 "" {HYPERLNK 17 "qgosper" 2 "qgospe r" "" }{TEXT 26 5 " and " }{HYPERLNK 17 "qsumrecursion" 2 "qsumrecursi on" "" }{TEXT 26 1 ":" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 112 "The most time consuming part in qgosper and qsumrecursion consist s of solving a system of linear equations (see " }{HYPERLNK 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 14 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 72 "sumdeltanabla, `sumdelta+nabla` - a variation of Zeilberger's algori thm" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 61 "\n sumdeltanable(f, k, p(x));\n `sumdelta+nabla`(f,k,p(x));" } }{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 4 "f - " }{TEXT -1 17 "an expression\n " }{TEXT 23 4 "k - " } {TEXT -1 34 "a name, the summation variable\n " }{TEXT 23 4 "x - " } {TEXT -1 35 "a name, the differibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctintdiffeqalgorithmalmkvistzeilbergdeterminholonomicdifferentialequatdefinitintegralusagcallsequencparameterexpressnameintegratvariablfunctisynopsidescriptimplementatlmkvisttriedeterminholonomicintsgxgintgfgtgagbgassumhyperexponentialwrtcontinuouverssumdiffeqtermggyperexponentialratiodiffdiffgtgfrationalalgorithmenhanccontgospcontgosperapplisumsigmaxgfsumgsigmjgthuscalculatantiderivatsigmagsuchoverwegetresultentialjgfbgffbwarnesrighthandsideabovrecurrencvanishreturnhomogeneouprocedurwilllookorderupmaxordglobalsetdefaultbutmaychanganypositintegvalupossiblobtaincertificathsumtheorallowsimplposterioproofusingthoptionalargumtruewitentrfirstcomputsecondexamplreadmplvpackaghypergeometricsummatycopyrightwolframkoepfuniversitkasselgconsidinfinitinfinitygfeulerrelatgam "WZcertificate" 2 "WZcertificate" "" }{TEXT -1 1 " " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 } PMATH 20 "6#>%\"GG,$*,,&%\"nG \"\"\"F)F)F),&%\"kGF)F)!\"\"F),(F(F,F)F,F+F)F,F+F)-%)binomialG6$F(F+F) F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simpcomb(rec-(subs(k= k+1,G)-G));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "gosper" 2 "gosper" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "hyperterm" 2 "hyperterm" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intrecursion" 2 "intrecursion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "kfreerec and fasenmyer" 2 "fasenmyer" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "sumdeltanabla" 2 "sumdeltanabla" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffeq" 2 "sumdiffeq" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "sumdiffrule and sumintrule" 2 "sumdiffrule" "" }{TEXT -1 2 ", " }{HYPERLNK 17^" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 259 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 | 1 "" {XPPMATH 20 "6#>%$re cG,&*(%\"nG\"\"\"%\"kGF(-%)binomialG6$,&F'F(F(F(F)F(!\"\"**\"\"#F(F-F( F)F(-F+6$F'F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG,$*,,&%\"nG \"\"\"F)F)F),&%\"kGF)F)!\"\"F),(F(F,F)F,F+F)F,F+F)-%)binomialG6$F(F+F) F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simpcomb(rec-(subs(k= k+1,G)-G));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "gosper" 2 "gosper" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "hyperterm" 2 "hyperterm" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intrecursion" 2 "intrecursion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "kfreerec and fasenmyer" 2 "fasenmyer" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "sumdeltanabla" 2 "sumdeltanabla" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffeq" 2 "sumdiffeq" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "sumdiffrule and sumintrule" 2 "sumdiffrule" "" }{TEXT -1 2 ", " }{HYPERLNK 17^*FGF*F/F*F*F)F*F*F:F*,&*&F+F*FGF*F/F)F*F*,(F)F*F*F*FGF/!\" #,(F)F*F+F*FGF/FM" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Calculate a \+ recurrence equation for " }{XPPEDIT 18 0 "s(n)=sum(k*binomial(n,k),k=0 ..n-2)" "6#/-%\"sG6#%\"nG-%$sumG6$*&%\"kG\"\"\"-%)binomialG6$F'F,F-/F, ;\"\"!,&F'F-\"\"#!\"\"" }{TEXT -1 1 ":" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "tmp:=sumrecursion(k*binomial(n,k),k=0..n-2,s(n), certificate=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tmpG7$/,&*&% \"nG\"\"\"-%\"sG6#,&F)F*F*F*F*!\"\"*(\"\"#F*F.F*-F,6#F)F*F*,$*(,&F)F*F *F/F*F)F*F.F*F/,$*(F.F*,&%\"kGF*F*F/F*,(F)F/F*F/F:F*F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Now prove the correspoding recurrence equ ation for the summand " }{XPPEDIT 18 0 "k*binomial(n,k)" "6#*&%\"kG\" \"\"-%)binomialG6$%\"nGF$F%" }{TEXT -1 6 " (see " }{HYPERLNK 17 "hsum[ theory]" 2 "hsum[theory]" "" }{TEXT -1 2 "):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "rec:= eval(subs(s=(n->k*binomial(n,k)),lhs(tmp[1]))); \nG:= tmp[2]*k*binomial(n,k);" }}{PARA 11 ""`&6%7%*&F.F-F)F-*&F. F-F*F-,&F)F-F*F-7$,(F)F-F*F-*&F-F-F.!\"\"F-,&*&F.F-F)F-F-*&F.F-F*F-F-F /" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Closedform(hyperterm([a,b],[a+ b+1/2],x,j)*\n hyperterm([a,b],[a+b+1/2],x,k-j),j,k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*HypertermG6&7%,$*&\"\"#\"\"\"%\"bGF *F*,$*&F)F*%\"aGF*F*,&F.F*F+F*7$,&*&F)F*F.F*F**&F)F*F+F*F*,(F.F*F+F*#F *F)F*%\"xG%\"kG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sumrecur sion(binomial(n,k)^3,k,f(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*& ),&%\"nG\"\"\"\"\"#F)F*F)-%\"fG6#F'F)!\"\"*&,(*&\"\"(F))F(F*F)F)*&\"#@ F)F(F)F)\"#;F)F)-F,6#,&F(F)F)F)F)F)*(\"\")F))F9F*F)-F,6#F(F)F)\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "sumrecursion(binomial(n,k) ^2*binomial(2*k,n),k,s(n),certificate=true,recorder=2);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7$/,(*&),&%\"nG\"\"\"\"\"#F*F+F*-%\"sG6#F(F*!\" \"*&,(*&\"\"(F*)F)F+F*F**&\"#@F*F)F*F*\"#;F*F*-F-6#,&F)F*F*F*F*F**(\" \")F*)F:F+F*-F-6#F)F*F*\"\"!*0,(*&\"\"$F*F)F*F*\"\"'F**&F+F*%\"kGF*F/F *FGF+,(*&F+Fa1 34 "a recurrence equation for the sum " }{XPPEDIT 18 0 "s(n)" "6#-%\"sG6#%\"nG" }{TEXT -1 25 " on the left hand side..." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "zeilberger((-1)^k*binomial(n+b,n+k) *binomial(n+c,c+k)*binomial(b+c,b+k),k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,*%\"bG\"\"\"%\"nGF(F(F(%\"cGF(F(-%\"sG6#F)F(F(*&, &F(!\"\"F)F0F(-F,6#,&F)F(F(F(F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "...and the corresponding right hand side determined by cl osedform:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "closedform((-1)^k*bino mial(n+b,n+k)*binomial(n+c,c+k)*binomial(b+c,b+k),k,n);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*(-%)binomialG6$,&%\"bG\"\"\"%\"cGF)F(F)-%+pochh ammerG6$,(F(F)F*F)F)F)%\"nGF)-%*factorialG6#F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "An application of Closedform to deduce Clausen' s formula:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "Hypergeom([a,b],[a+b+1, 2],x)^2=Hypergeom([2*a,2*b,a+b],[a+b+1/2,2*a+2*b],x)" "6#/*$-%*Hyperge omG6%7$%\"aG%\"bG7$,(F)\"\"\"F*F-F-F-\"\"#%\"xGF.-Fb 9 "Examples:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Mapl e~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-200 2,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Binomial Identity:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " zeilberger(binomial(n,k),k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,&*&\"\"#\"\"\"-%\"sG6#%\"nGF'F'-F)6#,&F+F'F'F'!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Dixon's identity," }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sum((-1)^k*binomial(n+b,n+k)*binomial(n+c,c+k)*binomial (b+c,b+k),k=-infinity..infinity)=GAMMA(b+c+n+1)/(n!*GAMMA(b+1)*GAMMA(c +1))" "6#/-%$sumG6$**),$\"\"\"!\"\"%\"kGF*-%)binomialG6$,&%\"nGF*%\"bG F*,&F1F*F,F*F*-F.6$,&F1F*%\"cGF*,&F7F*F,F*F*-F.6$,&F2F*F7F*,&F2F*F,F*F */F,;,$%)infinityGF+F@*&-%&GAMMAG6#,*F2F*F7F*F1F*F*F*F**(-%*factorialG 6#F1F*-FC6#,&F2F*F*F*F*-FC6#,&F7F*F*F*F*F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -c while sumrecursion searches for recurrence equation s of order one up to " }{TEXT 256 8 "MAXORDER" }{TEXT -1 312 ". This g lobal variable is set by default to 5 but may be changed to any positi ve integer value or infinity. For sumrecursion you can also use the op tional argument recorder=r..s or recorder =r with nonnegative integers r and to look for a recurrence equation in the range r..s or of order exactly r respectively." }}{PARA 15 "" 0 "" {TEXT -1 193 "The procedu res closedform and Closedform try to determine a recurrence equation o f first order via the procedure zeilberger. If successful this recurre nce equation is solved with initial value " }{XPPEDIT 18 0 "s(0)" "6#- %\"sG6#\"\"!" }{TEXT -1 100 " and the solution - a hypergeometric term - is returned: The procedure closedform uses the function " } {HYPERLNK 17 "hyperterm" 2 "hyperterm" "" }{TEXT -1 91 " to return the solution whereas the procedure Closedform uses the inert function Hyp erterm." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26d the summa tion bounds are " }{TEXT 258 11 "not natural" }{TEXT -1 162 ", one sho uld always use the call\n sumrecursion(f,k=a..b,s(n));\nthus enfo rcing the computation of the inhomogeneous part of the recurrence equa tion (see also " }{HYPERLNK 17 "hsum[theory]" 2 "hsum[theory]" "" } {TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXT -1 60 "With all these procedu res it is also possible to obtain the " }{TEXT 261 20 "rational certif icate" }{TEXT -1 6 " (see " }{HYPERLNK 17 "hsum[theory]" 2 "hsum[theor y]" "" }{TEXT -1 239 ") which allows a simple a posterio proof of the \+ calculated recurrence equation. When using the optional argument certi ficate=true, sumrecursion will return a list with two entries. The fir st is the computed recurrence equation for the sum " }{XPPEDIT 18 0 "s (n)" "6#-%\"sG6#%\"nG" }{TEXT -1 29 " and the second entry is the " } {TEXT 262 20 "rational certificate" }{TEXT -1 1 "." }}{PARA 15 "" 0 " " {TEXT -1 145 "Note that zeilberger looks only for a recurrence equat ion of first ordere"n" "6#%\"nG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 232 ". This is typically the cas e for ratios of products of rational functions, powers, factorials, bi nomial coefficients, and Pochhammer symbols that are integer-linear in their arguments. The implementation supports this type of input." }} {PARA 15 "" 0 "" {TEXT -1 84 "The output of sumrecursion (without any \+ summation bounds) or zeilberger is always a " }{TEXT 257 22 "homogeneo us recurrence" }{TEXT -1 26 ". I.e. it is assumed that " }{XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 49 " has finite support an d the summation bounds are " }{TEXT 259 7 "natural" }{TEXT -1 2 " (" } {XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 38 " is zero o utside the summation range):" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "Sum(f (n,k),k=a[n]..b[n])=Sum(f(n,k),k=-infinity..infinity)" "6#/-%$SumG6$-% \"fG6$%\"nG%\"kG/F+;&%\"aG6#F*&%\"bG6#F*-F%6$-F(6$F*F+/F+;,$%)infinity G!\"\"F;" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 28 "Iff bG6#F'" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 35 " is hyperg eometric with respect to " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 98 ", the functions close dform and Closedform try to determine a hypergeometric term that is eq ual to " }{XPPEDIT 18 0 "s(n)" "6#-%\"sG6#%\"nG" }{TEXT -1 1 "." }} {PARA 15 "" 0 "" {TEXT -1 14 "An expression " }{XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 11 " is called " }{TEXT 260 14 "hyper geometric" }{TEXT -1 23 " term with respect to (" }{XPPEDIT 18 0 "n" " 6#%\"nG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 4 ") i f" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "f(n+1,k)/f(n,k)" "6#*&-%\"fG6$,& %\"nG\"\"\"F)F)%\"kGF)-F%6$F(F*!\"\"" }{TEXT -1 22 " and \+ " }{XPPEDIT 18 0 "f(n,k+1)/f(n,k)" "6#*&-%\"fG6$%\"nG,&%\"kG\"\"\" F*F*F*-F%6$F'F)!\"\"" }}{PARA 14 "" 0 "" {TEXT -1 29 "are rational wit h respect to " }{XPPEDIT 18 0 g0 "" {TEXT 26 10 "Function: " }{TEXT -1 61 "closedform, sumrecursion, zeilberger - Zeilberger's algorithm" }} {PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 128 "\n closedform(f,k,n);\n Closedform(f,k,n);\n sumrecursion(f,k,s(n) );\n sumrecursion(f,k=a..b,s(n));\n zeilberger(f,k,s(n));" }} {PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 4 "f - " }{TEXT -1 17 "an expression\n " }{TEXT 23 4 "k - " } {TEXT -1 34 "a name, the summation variable\n " }{TEXT 23 4 "n - " } {TEXT -1 35 "a name, the recurrence variable\n " }{TEXT 23 4 "s - " }{TEXT -1 31 "a name, the recurrence function" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 148 "These functions are implementations of Zeilberger's algorithm. Wh ile sumrecursion and zeilberger calculate an upward recurrence equatio n for the sum" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "s(n)=sum(f(n,k),k=a[ n]..b[n])" "6#/-%\"sG6#%\"nG-%$sumG6$-%\"fG6$F'%\"kG/F.;&%\"aG6#F'&%\"hwolfwolfrwolframMworkworkshop writwrittenwrongwrt7wwwwz  wzcertifica wzcertificat/!xamplxgs    xgfcWxpresssumint  sumintrulGsummsummasummand?summatqsummatiosummin sumrecu sumrecur sumrecurskY     sumrecursisumssumtsumtohyp sumtool " 6#-&%&sigmaG6#%\"jG6#%\"xG" }{TEXT -1 11 " such that:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "f(x,t)+sum(sigma[j](x)*diff(f(x,t),`$`(x,j)),j = 1 .. J) = diff(G(x,t),t);" "6#/,&-%\"fG6$%\"xG%\"tG\"\"\"-%$sumG6$*&-&% &sigmaG6#%\"jG6#%\"xGF*-%%diffG6$-F&6$%\"xGF)-%\"$G6$F(F3F*/F3;F*%\"JG F*-F76$-%\"GG6$F(F)F)" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 34 "By integrating this equation over " }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT -1 6 " from " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 61 " we get the resulting differ ential equation for the integral " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#% \"xG" }{TEXT -1 1 ":" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "s(x)+sum(sigm a[j](x)*diff(s(x),`$`(x,j)),j = 1 .. J) = G(x,b)-G(x,a);" "6#/,&-%\"sG 6#%\"xG\"\"\"-%$sumG6$*&-&%&sigmaG6#%\"jG6#%\"xGF)-%%diffG6$-F&6#F(-% \"$G6$F(F2F)/F2;F)%\"JGF),&-%\"GG6$F(%\"bGF)-FB6$F(%\"aG!\"\"" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 134 "Warning: The algorithm assum es that the right-hand side oararameterarbitrar+ arcsinarcta argum3argumentS6aroundarrowgf  artificia ary asenmy asic#aske 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "B ullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 81 "contgosper - continuous analogue of Gosper's algorithm by Almkvist an d Zeilberger" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" } {TEXT -1 21 "\n contgosper(f, k);" }}{PARA 0 "" 0 "" {TEXT 26 11 "Pa rameters:" }{TEXT -1 4 "\n " }{TEXT 23 6 "f - " }{TEXT -1 17 "an e xpression\n " }{TEXT 23Q0{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 " " 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }ihomogeneouthoscaseneccessarcomputrecursisavingoftenreferinhomogeneourighthandsideecurrencguedecisprocedurdecidwhethhypergeometrictermfgkgantidifferencggreducproblemfindsuchantidifferencefinitlaurpolynomialsatisffirstorderrecurrencequatmainstepcalculatrepresentatrationalfunctpgrgwithiigcdgcdgjgfallnonnegatintegerjgxgsolvinhomogeneoudeterminlowerupperdegreboundsubstitutgenerichesystemlinearequationesultcomparcoefficientmonomialnosolutwecanconcludexisthypergometricnoteratiozeilbergdealdefnitsummathypergeomtricngrespecttrierecurrencsumsggivensumgagbgideadeducsummandapplysigmasigmagyetundeterminigmapositintegereasyshowalogudeliversystemabovthusinsteadonlyalsogetequationkgfmultiplreallysimplprovposterioriindependderivatjustdividverifresultcallcertificatequatisummoveralphaalphagmaxbetabetagminobtainlphaetafofdflsupportaturalinfinitinfinitygpartanishqc{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 " Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1geneous part v anishes, e.g. we get the homogeneous recurrence equation" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "sum(sigma[j](n)*S(n-j),j=0..J)=0" "6#/-%$sumG6$ *&-&%&sigmaG6#%\"jG6#%\"nG\"\"\"-%\"SG6#,&F.F/F,!\"\"F//F,;\"\"!%\"JGF 7" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 248 "I.e. in those cases it is not neccessary to compute the inhomogeneous part of the recursi on thus saving computing time. Note that we often refer to the inhomog eneous part of the recurrence equation as the right-hand side of the r ecurrence equation." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } 1 3 "If " }{XPPEDIT 18 0 "F(n,k)" "6#-%\"FG6$%\"nG%\"kG" }{TEXT -1 19 " has finite support" }{TEXT -1 43 " and the summation bounds are n atural, i.e." }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "sum(F(n,k),k=a[n]..b[ n])=sum(F(n,k),k=-infinity..infinity)" "6#/-%$sumG6$-%\"FG6$%\"nG%\"kG /F+;&%\"aG6#F*&%\"bG6#F*-F%6$-F(6$F*F+/F+;,$%)infinityG!\"\"F;" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 80 "the inhomosexpzgexpginfinitygbetawgwgfbatemanrepresationcgdghypergeomdgfhypergeomgxgfomgxgnhypertermkgcgfagfnsbetagirgconvertirgfseealsoalsointdiffeqrodriguesrecursrodriguesdiffeqsumdiffrulsumintrulsumdiffrulearameterexpressnameintegratvariablsynopsidescriptimplementattrierecurrenceintsgngintgfgtgagbgassuminghyperexponentialwrthypergometriccontinuouverssumrecurssumrecursionremembtermggratiodiffdiffgtgfrationalenhanccontgospapplifunctisumsigmasumgsigmagjgthuscalculatantiderivatsuchiffgoverwegetresultbgfwarnalgorithmrighthandsideabovequatiovanishreturnhomogeneouprocedurwilllookordeupmaxordglobalsetdefaultbutmaychanganypositintegvalueinfinitpossiblobtainrationalcertificathsumtheorallowsimplposterioproofusingoptionalargumtruewilwithentrfirstcomputsecondcertificatexamplreadhsumplvpackaghypergeometricsummatcopyrightwolframkoepfuniversitkasselggammagammagfunctiontnitygfngfseealintrecurssumdeltanablaintdiffeqsumdiffrulsumintrulhelpheadnormallistitembulletfunctsumdiffeqvariatzeilbergalgorithmfinddifferentialequationusagcallsequencparameterexpressnamesummatiovariabldifferentialsumunctsynopsidescriptimplementattriedetermindinarequatinfinitsgxgsumgfgkginfinitygassumhyperexponentialwrthypergeometrictermggratiodiffdiffgxgfrespectiverationalalgorithapplicatgospalsosumrecursthuscalculatfirststepsummandformsigmasigmagjgsumminoverallintegerwebtainiffgjgfprovidlimitrighthansideexistwithlimitgkgfhomogeneouordinaryprocedurwilllookorderupmaxordglobalsetdefaultbutmaychanganypositintegvalupossiblobtaincertificathsumtheorallowsimplposterioproofusingthoptionalargumtruereturnwitentrcomputsecondexamplreadmplvpackagsummatycopyrightwolframkoepfuniversitkasselgexpexpgactorialgsinsingfactorialglegendrpolynomialpgnggivebinomialbinomialginfiu 0 "> " 0 "" {MPLTEXT 1 0 63 " WZcertificate((-1)^k/(-3)^n*binomial(n,k)*binomial(3*k,n),k,n);" }} {PARA 8 "" 1 "" {TEXT -1 51 "Error, (in WZcertificate) Extended WZ met hod fails\n" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "go sper" 2 "gosper" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "kfreerec and fasen myer" 2 "fasenmyer" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "closedform, sum recursion and zeilberger" 2 "sumrecursion" "" }{TEXT -1 1 " " }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } kGF)F),&%\"bGF)F*F)F),*F, F)F(F)%\"nGF)F)F)F&,(F.F)F)F)F*F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Here is an example of a valid identity that cannot be pro ved by the WZ method:" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "sum((-1)^k*b inomial(n,k)*binomial(3*k,n),k=-infinity..infinity)=(-3)^n" "6#/-%$sum G6$*(),$\"\"\"!\"\"%\"kGF*-%)binomialG6$%\"nGF,F*-F.6$*&\"\"$F*F,F*F0F */F,;,$%)infinityGF+F8),$F4F+F0" }}{PARAvk)*binomial(b+c,b+k),k=-infinity..infinity)=GAMMA(b+c+n+1)/n!/GAMMA(b +1)/GAMMA(c+1)" "6#/-%$sumG6$**),$\"\"\"!\"\"%\"kGF*-%)binomialG6$,&% \"nGF*%\"bGF*,&F1F*F,F*F*-F.6$,&F1F*%\"cGF*,&F7F*F,F*F*-F.6$,&F2F*F7F* ,&F2F*F,F*F*/F,;,$%)infinityGF+F@**-%&GAMMAG6#,*F2F*F7F*F1F*F*F*F*-%*f actorialG6#F1F+-FC6#,&F2F*F*F*F+-FC6#,&F7F*F*F*F+" }}{PARA 0 "" 0 "" {TEXT -1 7 "yields:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "WZcertifica te((-1)^k*binomial(n+b,n+k)*binomial(n+c,c+k)*binomial(b+c,b+k)/(GAMMA (b+c+n+1)/(n!*GAMMA(b+1)*GAMMA(c+1))),k,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"#!\"\",&%\"cG\"\"\"%\"kGF)F),&%\"bGF)F*F)F),*F, F)F(F)%\"nGF)F)F)F&,(F.F)F)F)F*F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Here is an example of a valid identity that cannot be pro ved by the WZ method:" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "sum((-1)^k*b inomial(n,k)*binomial(3*k,n),k=-infinity..infinity)=(-3)^n" "6#/-%$sum G6$*(),$\"\"\"!\"\"%\"kGF*-%)binomialG6$%\"nGF,F*-F.6$*&\"\"$F*F,F*F0F */F,;,$%)infinityGF+F8),$F4F+F0" }}{PARAvnity)=1" "6#/-%$sumG6$*&-%\"FG6 $\"\"!%\"kG\"\"\"-%\"CG6#F+!\"\"/F,;,$%)infinityGF1F5F-" }{TEXT -1 1 " ." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Ma ple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfr am~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Let's prove the binomial identity" }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "sum(binomial(n,k)*x^k,k=0..n)=(1+x)^n" "6#/-%$sumG6$*&-%)binomia lG6$%\"nG%\"kG\"\"\")%\"xGF,F-/F,;\"\"!F+),&F-F-F/F-F+" }{TEXT -1 1 ": " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "WZcertificate(binomial(n,k)*x^k /(1+x)^n,k,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"kG\"\"\",(%\" nGF&F&F&F%!\"\"F),&F&F&%\"xGF&F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "An application of the WZ method to Dixon's identity," }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "sum((-1)^k*binomial(n+b,n+k)*binomial(n+c,c +w%\"nG\"\"\"F*F*F*-F&6#F)!\"\"\"\"!" }{TEXT -1 62 " is valid. If the method fails, an error message is returned." }}{PARA 15 "" 0 " " {TEXT -1 68 "Thus the WZ method may be successful to prove identitie s of the form" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "sum(F(n,k),k=-infini ty..infinity)=C(n)" "6#/-%$sumG6$-%\"FG6$%\"nG%\"kG/F+;,$%)infinityG! \"\"F/-%\"CG6#F*" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "C(n)" "6#-%\"CG6#%\"nG" }{TEXT -1 31 " is a hyperge ometric term wrt. " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 58 " not co ntaining sums with summation bounds that depend on " }{XPPEDIT 18 0 "n " "6#%\"nG" }{TEXT -1 59 ". All we have to do is apply the procedure W Zcertificate to" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "f(n,k)=F(n,k)/C(n) " "6#/-%\"fG6$%\"nG%\"kG*&-%\"FG6$F'F(\"\"\"-%\"CG6#F'!\"\"" }}{PARA 14 "" 0 "" {TEXT -1 13 "and check if " }{XPPEDIT 18 0 "s(0)=1" "6#/-% \"sG6#\"\"!\"\"\"" }{TEXT -1 6 ", i.e." }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "sum(F(0,k)/C(0),k=-infinity..infixulfils the following \+ equation:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "f(n+1,k)-f(n,k)=R(n,k+1) *f(n,k+1)-R(n,k)*f(n,k)" "6#/,&-%\"fG6$,&%\"nG\"\"\"F*F*%\"kGF*-F&6$F) F+!\"\",&*&-%\"RG6$F),&F+F*F*F*F*-F&6$F),&F+F*F*F*F*F**&-F26$F)F+F*-F& 6$F)F+F*F." }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 29 "By summing over all integers " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 30 " (usin g the finite support of " }{XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"k G" }{TEXT -1 47 " and some further knowledge about the poles of " } {XPPEDIT 18 0 "R(n,k)" "6#-%\"RG6$%\"nG%\"kG" }{TEXT -1 50 ") we get a simple recurrence equation for the sum " }{XPPEDIT 18 0 "s(n)" "6#-% \"sG6#%\"nG" }{TEXT -1 1 ":" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "s(n+1) -s(n)=0" "6#/,&-%\"sG6#,&%\"nG\"\"\"F*F*F*-F&6#F)!\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 39 "So if WZcertificate returns a function " }{XPPEDIT 18 0 "R(n,k)" "6#-%\"RG6$%\"nG%\"kG" }{TEXT -1 26 " the recurrence equation " }{XPPEDIT 18 0 "s(n+1)-s(n)=0" "6#/,&- %\"sG6#,&y 2 " (" }{TEXT 256 9 "WZ method" }{TEXT -1 51 ") that can be used to prove identities of the form:" }}{PARA 256 "" 0 " " {XPPEDIT 18 0 "s(n)=1" "6#/-%\"sG6#%\"nG\"\"\"" }{TEXT -1 15 ", \+ where " }{XPPEDIT 18 0 "s(n)=sum(f(n,k),k=-infinity..infinity)" "6# /-%\"sG6#%\"nG-%$sumG6$-%\"fG6$F'%\"kG/F.;,$%)infinityG!\"\"F2" }} {PARA 14 "" 0 "" {TEXT -1 14 "assuming that " }{XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 51 " is a hypergeometric term with fi nite support wrt. " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 7 ", i.e. \+ " }{XPPEDIT 18 0 "f(n,k)" "6#-%\"fG6$%\"nG%\"kG" }{TEXT -1 48 " is ide ntical zero outside a finite interval in " }{XPPEDIT 18 0 "k" "6#%\"kG " }{TEXT -1 10 " for each " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 1 " ." }}{PARA 15 "" 0 "" {TEXT -1 55 "If successful the function returns \+ a rational function " }{XPPEDIT 18 0 "R(n,k)" "6#-%\"RG6$%\"nG%\"kG" } {TEXT -1 4 " in " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 37 " that fz0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 38 "WZcertificate - Wilf-Zeilberger method" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 25 "\n WZcertificate(f,k,n);" }} {PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 4 "f - " }{TEXT -1 17 "an expression\n " }{TEXT 23 4 "k - " } {TEXT -1 34 "a name, the summation variable\n " }{TEXT 23 4 "n - " } {TEXT -1 31 "a name, the recurrence variable" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 42 " This function is an implementation of the " }{TEXT 257 22 "Wilf-Zeilbe rger method" }{TEXT -1{*FGF*F/F*F*F)F*F*F:F*,&*&F+F*FGF*F/F)F*F*,(F)F*F*F*FGF/!\" #,(F)F*F+F*FGF/FM" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Calculate a \+ recurrence equation for " }{XPPEDIT 18 0 "s(n)=sum(k*binomial(n,k),k=0 ..n-2)" "6#/-%\"sG6#%\"nG-%$sumG6$*&%\"kG\"\"\"-%)binomialG6$F'F,F-/F, ;\"\"!,&F'F-\"\"#!\"\"" }{TEXT -1 1 ":" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "tmp:=sumrecursion(k*binomial(n,k),k=0..n-2,s(n), certificate=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tmpG7$/,&*&% \"nG\"\"\"-%\"sG6#,&F)F*F*F*F*!\"\"*(\"\"#F*F.F*-F,6#F)F*F*,$*(,&F)F*F *F/F*F)F*F.F*F/,$*(F.F*,&%\"kGF*F*F/F*,(F)F/F*F/F:F*F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Now prove the correspoding recurrence equ ation for the summand " }{XPPEDIT 18 0 "k*binomial(n,k)" "6#*&%\"kG\" \"\"-%)binomialG6$%\"nGF$F%" }{TEXT -1 6 " (see " }{HYPERLNK 17 "hsum[ theory]" 2 "hsum[theory]" "" }{TEXT -1 2 "):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "rec:= eval(subs(s=(n->k*binomial(n,k)),lhs(tmp[1]))); \nG:= tmp[2]*k*binomial(n,k);" }}{PARA 11 ""iQ0{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 " " 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } "false" }{TEXT -1 186 ". I.e. the most general solution that might include some arbitrar y constants is returned. Those constants have the names _C1, _C2, ... \nIf set to true those constants will be set to zero." }}}{SECT 0 {PARA 4 "" 0 "" {HYPERLNK 17 "qgosper" 2 "qgosper" "" }{TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 20 "The global variable " }{TEXT 19 23 "_q gosper_antidifference" }{TEXT -1 15 " may be set to " }{TEXT 19 4 "dow n" }{TEXT -1 4 " or " }{TEXT 19 2 "up" }{TEXT -1 13 ", default is " } {TEXT 19 2 "up" }{TEXT -1 52 ".\nThe values down or up mean that qgosp er returns a " }{TEXT 260 8 "downward" }{TEXT -1 4 " or " }{TEXT 261 6 "upward" }{TEXT -1 1 " " }{TEXT 262 14 "antidifference" }{TEXT -1 15 ". The function " }{XPPEDIT 18 0 "g(k)" "6#-%\"gG6#%\"kG" }{TEXT -1 7 " is an " }{TEXT 263 6 "upward" }{TEXT -1 4 " or " }{TEXT 264 8 " downward" }{TEXT -1 1 " " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 1 "- " }{TEXT 265 29 "hypergeometric antidifference" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "f(k)N2"{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warni ng" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 " _rterm" 2 "hyperterm " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intrecursion" 2 "intrecursion" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffeq" 2 "sumdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffrule and sumintrule" 2 "sumdiffrule" "" }{TEXT -1 1 " " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 } F)F/F)FSF)F )F+F)F:F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "`sumdel ta+nabla`(wilsonterm,k,S(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*. ,&*&\"\"#\"\"\"%\"xGF)F)F)!\"\"F),&F*F)%\"cGF)F),&%\"dGF)F*F)F),&%\"bG F)F*F)F),&%\"aGF)F*F)F)-%&DeltaG6$-%\"SG6#F*F*F)F+*.,&*&F(F)F*F)F)F)F) F),&F*F)F-F+F),&F*F)F/F+F),&F*F)F1F+F),&F3F)F*F+F)-%&NablaGF6F)F+*0F(F )%\"nGF)F*F)F&F)F;F),.F3F)F1F)F-F)F/F)FDF)F)F+F)F7F)F)\"\"!" }}}} {SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumrecursion" 2 "su mrecursion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "hypec{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 " Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1rtificate" 2 "WZcertificate" "" }{TEXT -1 1 " " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ,&*&F(F )F,F)F)\"\"%F)F),&*&F(F)F,F)F)\"\")F)F)-%\"sG6#,&F,F)F(F)F)F)**F0F)F.F ),(*&\"#PF))F,\"\"#F)F)*&\"$!=F)F,F)F)\"$=#F)F)-F56#,&F,F)F=F)F)F)**\" #;F)FCF),(*&\"#LF)F " 0 "" {MPLTEXT 1 0 18 "rechyper(re,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<#!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "rechyper((n+4)*s (n+2)+s(n+1)-(n+1)*s(n),s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,$ **,&*&\"\"#\"\"\"%\"nGF)F)\"\"&F)F),&F*F)F)F)F),&*&F(F)F*F)F)\"\"$F)! \"\",&F*F)F/F)F0F0*&F,F)F1F0" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "kfreerec and fasenmyer" 2 "fasenmyer" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "closedform and sumrecursion" 2 "sumrecursion " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "WZceon((-1)^k*binomial(n,k)*binomial(4*k,n),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#reG/,**,\"\"$\"\"\",&*&F(F)%\"nGF)F)\"\"(F)F),&*&F(F )F,F)F)\"\"%F)F),&*&F(F)F,F)F)\"\")F)F)-%\"sG6#,&F,F)F(F)F)F)**F0F)F.F ),(*&\"#PF))F,\"\"#F)F)*&\"$!=F)F,F)F)\"$=#F)F)-F56#,&F,F)F=F)F)F)**\" #;F)FCF),(*&\"#LF)F " 0 "" {MPLTEXT 1 0 18 "rechyper(re,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<#!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "rechyper((n+4)*s (n+2)+s(n+1)-(n+1)*s(n),s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,$ **,&*&\"\"#\"\"\"%\"nGF)F)\"\"&F)F),&F*F)F)F)F),&*&F(F)F*F)F)\"\"$F)! \"\",&F*F)F/F)F0F0*&F,F)F1F0" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "kfreerec and fasenmyer" 2 "fasenmyer" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "closedform and sumrecursion" 2 "sumrecursion " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "WZcere the WZ-method failed (see " } {HYPERLNK 17 "WZcertificate" 2 "WZcertificate" "" }{TEXT -1 49 ") and \+ sumrecursion returns a recurrence of order " }{XPPEDIT 18 0 "d" "6#%\" dG" }{TEXT -1 26 " that is not minimal (for " }{XPPEDIT 18 0 "d>1" "6# 2\"\"\"%\"dG" }{TEXT -1 4 ") as" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "su m((-1)^k*binomial(n,k)*binomial(d*k,n),k=0..n)=(-d)^n" "6#/-%$sumG6$*( ),$\"\"\"!\"\"%\"kGF*-%)binomialG6$%\"nGF,F*-F.6$*&%\"dGF*F,F*F0F*/F,; \"\"!F0),$F4F+F0" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "for a ll positive integers d." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "re:= sum recursion((-1)^k*binomial(n,k)*binomial(3*k,n),k,s(n));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#reG/,(*(\"\"#\"\"\",&*&F(F)%\"nGF)F)\"\"$F)F) -%\"sG6#,&F,F)F(F)F)F)*(F-F),&*&\"\"&F)F,F)F)\"\"(F)F)-F/6#,&F,F)F)F)F )F)*(\"\"*F)F9F)-F/6#F,F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rec2hyper(re,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#<#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "re:= sumrecursi elt aG6#\"\"\"" }{TEXT -1 90 ",... they can be considered as arbitrary con stants. (Note that those variables are local!)" }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfram~Koepf,~Univer sity~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "rec2poly (n*(n+1)*s(n+2)-2*n*(n+10)*s(n+1)+(n+9)*(n+10)*s(n),s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*6%\"nG\"\"\",&F$F%\"\")F%F%,&F$F%\"\"(F%F%, &F$F%\"\"'F%F%,&F$F%\"\"&F%F%,&F$F%\"\"%F%F%,&F$F%\"\"$F%F%,&F$F%\"\"# F%F%,&F$F%F%F%F%,(*&F$F%&%&deltaG6#\"\"!F%F%&F86#F%F%*&\"#OF%F7F%!\"\" F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "rec2poly(n*s(n+2)-2*n *s(n+1)+(n+10)*s(n),s(n));" }}{PARA 8 "" 1 "" {TEXT -1 51 "Error, (in \+ rec2poly) No polynomial solution exists\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Here an example whe polynomial/hypergeometric ter m solutions of a given homogeneous recurrence equation with polynomial coefficients in " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 135 ". The c orresponding procedures rec2poly and rec2hyper are special versions th at can only be used for recurrence equations of order two." }}{PARA 15 "" 0 "" {TEXT -1 125 "The output of the procedures is a set contain ing all possible solutions. I.e. if no solution exists an empty set is returned." }}{PARA 15 "" 0 "" {TEXT -1 13 "Note that if " }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#%\"nG" }{TEXT -1 94 " is a solution of the recu rrence equation the procedures rec2hyper and rechyper do NOT return " }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#%\"nG" }{TEXT -1 31 " but rather the \+ rational ratio " }{XPPEDIT 18 0 "f(n+1)/f(n)" "6#*&-%\"fG6#,&%\"nG\"\" \"F)F)F)-F%6#F(!\"\"" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 39 " If the solution contains any variables " }{XPPEDIT 18 0 "delta[0]" "6# &%&deltaG6#\"\"!" }{TEXT -1 1 "," }{XPPEDIT 18 0 "delta[1]" "6#&%&d {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 41 "recpoly, rechyper - Petkovsek's algorithm" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 88 "\n rec2poly(re,s(n)); \n recpoly(re,s(n));\n rec2hyper(re,s(n));\n rechyper(re,s(n)); " }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " } {TEXT 23 5 "re - " }{TEXT -1 83 "an equation or algebraic expression, \+ the recurrence equation in s(n),s(n+1),...\n " }{TEXT 23 4 "n - " } {TEXT -1 35 "a name, the recurrence variable\n " }{TEXT 23 4 "s - " }{TEXT -1 31 "a name, the recurrence function" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 222 "These functions are implementations of Petkovsek's algorithm. The procedure recpoly/rechyper searches foroibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalerrorbulletitemnormalfunctrecporechyppetkovsekalgorithmusagcallsequencrecpolyrehyperparameterequatalgebraicexpressrecurrencnamevariablsynopsidescriptthesfunctionimplementatprocedursearchpolynomialhypergeometrictersolutiongivenhomogeneouwithcoefficientngorrespondspecialversionthatcanonlyusedequationordersetcontainingallpossiblnosolutexistemptreturnnotefgrecurrencdobutrathrationalratioanydeltadeltagdeltagtheyconsiderarbitrarconstantthoslocalexamplreadhsummplvpackagsummatycopyrightwolframkoepfunivsitykasselgherewzmethodfailwzcertificatsumrecursdgminimalsubinomialsumgkgfbinomialgngfdgfllpositintegersumrecursregsgsumrecursipffcflfkffnfseealsoalsokfreerecfasenmyclosedform#{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYx(ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalasicqgfunctioncallsequencqpochhamminfinitnqpochhammnqfacnqbracketnqfactorialnqbinomialdescriptqpochhammerfuncttakeargumentfirstparametanyalgebraicexpresssecondtypenameraisotintegerpowerthirdvariablalgebraicrepresingintegvalusymbolicdefinproductqpochhammergaginfinitygproductgjgfergkgpiecewisegproductgqfacunctsynonymqbracketbothcanvaliddefinitqbracketsgqbinomialusesexpressionsrepresentlastqbinomialgngqfactorialqfactorialqfactorialgqgammatwzgexamplreadqsummplppackaghypergeometricsummatcocopyrightharaldboeingwolframkoepuniversitkasselgrocedurqsimpcombprovidsimplificatmechanismallthosunctionagfqgfalsoqhypertermqpsihypertermqE{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Help H eading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 " " 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE"> " 0 " " {MPLTEXT 1 0 41 "normal(eval(subs(B(x)=BesselI(n,x),DE)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "seealso" {TEXT -1 10 "See \+ Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "FPS[FormalPowerSeries]" 2 "FPS [FormalPowerSeries]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "FPS[SimpleRE] " 2 "FPS[SimpleRE]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "FPS[standardsum ]." 2 "FPS[standardsum]" "" }}}}{MARK "13 1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } 1 0 31 "HolonomicDE(arcsin(x)^2, F(x));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,(*(,&%\"xG\"\"\"F(!\"\"F(,&F'F(F(F(F(-%%diffG6$-%\" FG6#F'-%\"$G6$F'\"\"$F(F(-F,6$F.F'F(*(F4F(F'F(-F,6$F.-F26$F'\"\"#F(F( \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "DE := HolonomicDE( BesselI(n,x), B(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/,(*&-%% diffG6$-%\"BG6#%\"xG-%\"$G6$F.\"\"#\"\"\")F.F2F3F3*&,&*$F4F3!\"\"*$)% \"nGF2F3F8F3F+F3F3*&-F)6$F+F.F3F.F3F3\"\"!" }}}{EXCHG {PARA 0 {VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "Bullet Item" -1 15 1 7-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qsimplify(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" }{TEXT 256 1 " " } {HYPERLNK 17 "qbrackets,qpochhammer,qfactorial, qGAMMA and qbinomial" 2 "qfunctions" "" }{TEXT 260 2 ", " }{HYPERLNK 17 "qhyperterm and qpsi hyperterm" 2 "qhyperterm" "" }{TEXT 259 2 ", " }{HYPERLNK 17 "qsimplif y" 2 "qsimplify" "" }{TEXT 261 2 ", " }{HYPERLNK 17 "qsum" 2 "qsum" " " }{TEXT 258 8 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ochha mmer(a,q^2,n)*\nqpochhammer(a*q,q^2,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(-%,qpochhammerG6%%\"aG%\"qG,$*&\"\"#\"\"\"%\"nGF.F.!\"\" -F'6%F)*$)F*F-F.F/F.-F'6%*&F)F.F*F.F3F/F." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "qsimpcomb(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*( -%,qpochhammerG6%%\"aG%\"qG,$*&\"\"#\"\"\"%\"nGF,F,!\"\"-F%6%F'*$)F(F+ F,F-F,-F%6%*&F'F,F(F,F1F&NablaG6$-%\"SG6#F*F*F*F)F+*(F(F),B*(F/F)F-F))F*F( F)F+*(F-F)F1F)F@F)F+*(F/F)F1F)F@F)F+*(F(F)F/F))F*\"\"%F)F)*(F(F)F3F)FD F)F)*$FDF)F+*(F(F)F-F)FDF)F)*,F(F)F/F)F-F)F1F)F@F)F)*(F(F)F1F)FDF)F)*, F(F)F/F)F3F)F1F)F@F)F)*,F(F)F-F)F3F)F1F)F@F)F)*(F3F)F1F)F@F)F+*,F(F)F/ F)F-F)F3F)F@F)F)*(F3F)F-F)F@F)F+**F3F)F/F)F-F)F1F)F+*(F3F)F/F)F@F)F+F) F7F)F+*0F(F)%\"nGF)F*F)F&F),&*&F(F)F*F)F)F)F)F),.F3F)F1F)F-F)F/F)FSF)F )F+F)F:F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "`sumdel ta+nabla`(wilsonterm,k,S(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*. ,&*&\"\"#\"\"\"%\"xGF)F)F)!\"\"F),&F*F)%\"cGF)F),&%\"dGF)F*F)F),&%\"bG F)F*F)F),&%\"aGF)F*F)F)-%&DeltaG6$-%\"SG6#F*F*F)F+*.,&*&F(F)F*F)F)F)F) F),&F*F)F-F+F),&F*F)F/F+F),&F*F)F1F+F),&F3F)F*F+F)-%&NablaGF6F)F+*0F(F )%\"nGF)F*F)F&F)F;F),.F3F)F1F)F-F)F/F)FDF)F)F+F)F7F)F)\"\"!" }}}} {SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumrecursion" 2 "su mrecursion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "hypertificate" 2 "WZcertificate" "" }{TEXT -1 1 " " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ,&*&F(F )F,F)F)\"\"%F)F),&*&F(F)F,F)F)\"\")F)F)-%\"sG6#,&F,F)F(F)F)F)**F0F)F.F ),(*&\"#PF))F,\"\"#F)F)*&\"$!=F)F,F)F)\"$=#F)F)-F56#,&F,F)F=F)F)F)**\" #;F)FCF),(*&\"#LF)F " 0 "" {MPLTEXT 1 0 18 "rechyper(re,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<#!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "rechyper((n+4)*s (n+2)+s(n+1)-(n+1)*s(n),s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,$ **,&*&\"\"#\"\"\"%\"nGF)F)\"\"&F)F),&F*F)F)F)F),&*&F(F)F*F)F)\"\"$F)! \"\",&F*F)F/F)F0F0*&F,F)F1F0" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "kfreerec and fasenmyer" 2 "fasenmyer" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "closedform and sumrecursion" 2 "sumrecursion " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "WZcertificate" 2 "WZcertificate" "" }{TEXT -1 1 " " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ,&*&F(F )F,F)F)\"\"%F)F),&*&F(F)F,F)F)\"\")F)F)-%\"sG6#,&F,F)F(F)F)F)**F0F)F.F ),(*&\"#PF))F,\"\"#F)F)*&\"$!=F)F,F)F)\"$=#F)F)-F56#,&F,F)F=F)F)F)**\" #;F)FCF),(*&\"#LF)F " 0 "" {MPLTEXT 1 0 18 "rechyper(re,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<#!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "rechyper((n+4)*s (n+2)+s(n+1)-(n+1)*s(n),s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,$ **,&*&\"\"#\"\"\"%\"nGF)F)\"\"&F)F),&F*F)F)F)F),&*&F(F)F*F)F)\"\"$F)! \"\",&F*F)F/F)F0F0*&F,F)F1F0" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "kfreerec and fasenmyer" 2 "fasenmyer" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "closedform and sumrecursion" 2 "sumrecursion " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "WZcec{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 " Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1on((-1)^k*binomial(n,k)*binomial(4*k,n),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#reG/,**,\"\"$\"\"\",&*&F(F)%\"nGF)F)\"\"(F)F),&*&F(F )F,F)F)\"\"%F)F),&*&F(F)F,F)F)\"\")F)F)-%\"sG6#,&F,F)F(F)F)F)**F0F)F.F ),(*&\"#PF))F,\"\"#F)F)*&\"$!=F)F,F)F)\"$=#F)F)-F56#,&F,F)F=F)F)F)**\" #;F)FCF),(*&\"#LF)F " 0 "" {MPLTEXT 1 0 18 "rechyper(re,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<#!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "rechyper((n+4)*s (n+2)+s(n+1)-(n+1)*s(n),s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,$ **,&*&\"\"#\"\"\"%\"nGF)F)\"\"&F)F),&F*F)F)F)F),&*&F(F)F*F)F)\"\"$F)! \"\",&F*F)F/F)F0F0*&F,F)F1F0" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "kfreerec and fasenmyer" 2 "fasenmyer" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "closedform and sumrecursion" 2 "sumrecursion " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "WZcere the WZ-method failed (see " } {HYPERLNK 17 "WZcertificate" 2 "WZcertificate" "" }{TEXT -1 49 ") and \+ sumrecursion returns a recurrence of order " }{XPPEDIT 18 0 "d" "6#%\" dG" }{TEXT -1 26 " that is not minimal (for " }{XPPEDIT 18 0 "d>1" "6# 2\"\"\"%\"dG" }{TEXT -1 4 ") as" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "su m((-1)^k*binomial(n,k)*binomial(d*k,n),k=0..n)=(-d)^n" "6#/-%$sumG6$*( ),$\"\"\"!\"\"%\"kGF*-%)binomialG6$%\"nGF,F*-F.6$*&%\"dGF*F,F*F0F*/F,; \"\"!F0),$F4F+F0" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "for a ll positive integers d." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "re:= sum recursion((-1)^k*binomial(n,k)*binomial(3*k,n),k,s(n));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#reG/,(*(\"\"#\"\"\",&*&F(F)%\"nGF)F)\"\"$F)F) -%\"sG6#,&F,F)F(F)F)F)*(F-F),&*&\"\"&F)F,F)F)\"\"(F)F)-F/6#,&F,F)F)F)F )F)*(\"\"*F)F9F)-F/6#F,F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rec2hyper(re,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#<#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "re:= sumrecursi elt aG6#\"\"\"" }{TEXT -1 90 ",... they can be considered as arbitrary con stants. (Note that those variables are local!)" }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfram~Koepf,~Univer sity~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "rec2poly (n*(n+1)*s(n+2)-2*n*(n+10)*s(n+1)+(n+9)*(n+10)*s(n),s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*6%\"nG\"\"\",&F$F%\"\")F%F%,&F$F%\"\"(F%F%, &F$F%\"\"'F%F%,&F$F%\"\"&F%F%,&F$F%\"\"%F%F%,&F$F%\"\"$F%F%,&F$F%\"\"# F%F%,&F$F%F%F%F%,(*&F$F%&%&deltaG6#\"\"!F%F%&F86#F%F%*&\"#OF%F7F%!\"\" F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "rec2poly(n*s(n+2)-2*n *s(n+1)+(n+10)*s(n),s(n));" }}{PARA 8 "" 1 "" {TEXT -1 51 "Error, (in \+ rec2poly) No polynomial solution exists\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Here an example whe polynomial/hypergeometric ter m solutions of a given homogeneous recurrence equation with polynomial coefficients in " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 135 ". The c orresponding procedures rec2poly and rec2hyper are special versions th at can only be used for recurrence equations of order two." }}{PARA 15 "" 0 "" {TEXT -1 125 "The output of the procedures is a set contain ing all possible solutions. I.e. if no solution exists an empty set is returned." }}{PARA 15 "" 0 "" {TEXT -1 13 "Note that if " }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#%\"nG" }{TEXT -1 94 " is a solution of the recu rrence equation the procedures rec2hyper and rechyper do NOT return " }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#%\"nG" }{TEXT -1 31 " but rather the \+ rational ratio " }{XPPEDIT 18 0 "f(n+1)/f(n)" "6#*&-%\"fG6#,&%\"nG\"\" \"F)F)F)-F%6#F(!\"\"" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 39 " If the solution contains any variables " }{XPPEDIT 18 0 "delta[0]" "6# &%&deltaG6#\"\"!" }{TEXT -1 1 "," }{XPPEDIT 18 0 "delta[1]" "6#&%&d {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 41 "recpoly, rechyper - Petkovsek's algorithm" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 88 "\n rec2poly(re,s(n)); \n recpoly(re,s(n));\n rec2hyper(re,s(n));\n rechyper(re,s(n)); " }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " } {TEXT 23 5 "re - " }{TEXT -1 83 "an equation or algebraic expression, \+ the recurrence equation in s(n),s(n+1),...\n " }{TEXT 23 4 "n - " } {TEXT -1 35 "a name, the recurrence variable\n " }{TEXT 23 4 "s - " }{TEXT -1 31 "a name, the recurrence function" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 222 "These functions are implementations of Petkovsek's algorithm. The procedure recpoly/rechyper searches foroibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalerrorbulletitemnormalfunctrecporechyppetkovsekalgorithmusagcallsequencrecpolyrehyperparameterequatalgebraicexpressrecurrencnamevariablsynopsidescriptthesfunctionimplementatprocedursearchpolynomialhypergeometrictersolutiongivenhomogeneouwithcoefficientngorrespondspecialversionthatcanonlyusedequationordersetcontainingallpossiblnosolutexistemptreturnnotefgrecurrencdobutrathrationalratioanydeltadeltagdeltagtheyconsiderarbitrarconstantthoslocalexamplreadhsummplvpackagsummatycopyrightwolframkoepfunivsitykasselgherewzmethodfailwzcertificatsumrecursdgminimalsubinomialsumgkgfbinomialgngfdgfllpositintegersumrecursregsgsumrecursipffcflfkffnfseealsoalsokfreerecfasenmyclosedform:ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctsumdiffeqvariatzeilbergalgorithmfinddifferentialequationusagcallsequencparameterexpressnamesummatiovariabldifferentialsumunctsynopsidescriptimplementattriedetermindinarequatinfinitsgxgsumgfgkginfinitygassumhyperexponentialwrthypergeometrictermggratiodiffdiffgxgfrespectiverationalalgorithapplicatgospalsosumrecursthuscalculatfirststepsummandformsigmasigmagjgsumminoverallintegerwebtainiffgjgfprovidlimitrighthansideexistwithlimitgkgfhomogeneouordinaryprocedurwilllookorderupmaxordglobalsetdefaultbutmaychanganypositintegvalupossiblobtaincertificathsumtheorallowsimplposterioproofusingthoptionalargumtruereturnwitentrcomputsecondexamplreadmplvpackagsummatycopyrightwolframkoepfuniversitkasselgexpexpgactorialgsinsingfactorialglegendrpolynomialpgnggivebinomialbinomialginfiuibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctsumdiffrulsumintrulvariatzeilbergalgorithmfindderivatintegratruleusagcallingsequencsumintparameterexpressnamesummatvariabldifferentialrecurrencsynopsidescriptletsuminfinitsgngxgsumgfgkgfinfinitygprocedurtriedeterminformdiffsigmadiffgxgfsigmagjgglobalmaxordgivemaximumshiftjgfcanappearrighthandsidedefaultsetbutyouassignanypositintegrocedurintegralintgigmagexamplreadhsummplvpackaghypergeometricycopyrightwolframkoepfuniversitkasselglegendrpolynomialbinomialbinomialpgbinomialgkginfinitygfbinomialntgseealalsocontgospintrecursintdiffeqgospsumdiffeqsumdeltanabladeltanablasumrecurszeilbrgerrtificate" 2 "WZcertificate" "" }{TEXT -1 1 " " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ,&*&F(F )F,F)F)\"\"%F)F),&*&F(F)F,F)F)\"\")F)F)-%\"sG6#,&F,F)F(F)F)F)**F0F)F.F ),(*&\"#PF))F,\"\"#F)F)*&\"$!=F)F,F)F)\"$=#F)F)-F56#,&F,F)F=F)F)F)**\" #;F)FCF),(*&\"#LF)F " 0 "" {MPLTEXT 1 0 18 "rechyper(re,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<#!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "rechyper((n+4)*s (n+2)+s(n+1)-(n+1)*s(n),s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,$ **,&*&\"\"#\"\"\"%\"nGF)F)\"\"&F)F),&F*F)F)F)F),&*&F(F)F*F)F)\"\"$F)! \"\",&F*F)F/F)F0F0*&F,F)F1F0" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "kfreerec and fasenmyer" 2 "fasenmyer" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "closedform and sumrecursion" 2 "sumrecursion " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "WZcec{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 " Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1on((-1)^k*binomial(n,k)*binomial(4*k,n),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#reG/,**,\"\"$\"\"\",&*&F(F)%\"nGF)F)\"\"(F)F),&*&F(F )F,F)F)\"\"%F)F),&*&F(F)F,F)F)\"\")F)F)-%\"sG6#,&F,F)F(F)F)F)**F0F)F.F ),(*&\"#PF))F,\"\"#F)F)*&\"$!=F)F,F)F)\"$=#F)F)-F56#,&F,F)F=F)F)F)**\" #;F)FCF),(*&\"#LF)F " 0 "" {MPLTEXT 1 0 18 "rechyper(re,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<#!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "rechyper((n+4)*s (n+2)+s(n+1)-(n+1)*s(n),s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,$ **,&*&\"\"#\"\"\"%\"nGF)F)\"\"&F)F),&F*F)F)F)F),&*&F(F)F*F)F)\"\"$F)! \"\",&F*F)F/F)F0F0*&F,F)F1F0" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "kfreerec and fasenmyer" 2 "fasenmyer" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "closedform and sumrecursion" 2 "sumrecursion " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "WZcere the WZ-method failed (see " } {HYPERLNK 17 "WZcertificate" 2 "WZcertificate" "" }{TEXT -1 49 ") and \+ sumrecursion returns a recurrence of order " }{XPPEDIT 18 0 "d" "6#%\" dG" }{TEXT -1 26 " that is not minimal (for " }{XPPEDIT 18 0 "d>1" "6# 2\"\"\"%\"dG" }{TEXT -1 4 ") as" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "su m((-1)^k*binomial(n,k)*binomial(d*k,n),k=0..n)=(-d)^n" "6#/-%$sumG6$*( ),$\"\"\"!\"\"%\"kGF*-%)binomialG6$%\"nGF,F*-F.6$*&%\"dGF*F,F*F0F*/F,; \"\"!F0),$F4F+F0" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "for a ll positive integers d." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "re:= sum recursion((-1)^k*binomial(n,k)*binomial(3*k,n),k,s(n));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#reG/,(*(\"\"#\"\"\",&*&F(F)%\"nGF)F)\"\"$F)F) -%\"sG6#,&F,F)F(F)F)F)*(F-F),&*&\"\"&F)F,F)F)\"\"(F)F)-F/6#,&F,F)F)F)F )F)*(\"\"*F)F9F)-F/6#F,F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rec2hyper(re,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#<#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "re:= sumrecursi elt aG6#\"\"\"" }{TEXT -1 90 ",... they can be considered as arbitrary con stants. (Note that those variables are local!)" }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfram~Koepf,~Univer sity~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "rec2poly (n*(n+1)*s(n+2)-2*n*(n+10)*s(n+1)+(n+9)*(n+10)*s(n),s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*6%\"nG\"\"\",&F$F%\"\")F%F%,&F$F%\"\"(F%F%, &F$F%\"\"'F%F%,&F$F%\"\"&F%F%,&F$F%\"\"%F%F%,&F$F%\"\"$F%F%,&F$F%\"\"# F%F%,&F$F%F%F%F%,(*&F$F%&%&deltaG6#\"\"!F%F%&F86#F%F%*&\"#OF%F7F%!\"\" F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "rec2poly(n*s(n+2)-2*n *s(n+1)+(n+10)*s(n),s(n));" }}{PARA 8 "" 1 "" {TEXT -1 51 "Error, (in \+ rec2poly) No polynomial solution exists\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Here an example whe polynomial/hypergeometric ter m solutions of a given homogeneous recurrence equation with polynomial coefficients in " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 135 ". The c orresponding procedures rec2poly and rec2hyper are special versions th at can only be used for recurrence equations of order two." }}{PARA 15 "" 0 "" {TEXT -1 125 "The output of the procedures is a set contain ing all possible solutions. I.e. if no solution exists an empty set is returned." }}{PARA 15 "" 0 "" {TEXT -1 13 "Note that if " }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#%\"nG" }{TEXT -1 94 " is a solution of the recu rrence equation the procedures rec2hyper and rechyper do NOT return " }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#%\"nG" }{TEXT -1 31 " but rather the \+ rational ratio " }{XPPEDIT 18 0 "f(n+1)/f(n)" "6#*&-%\"fG6#,&%\"nG\"\" \"F)F)F)-F%6#F(!\"\"" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 39 " If the solution contains any variables " }{XPPEDIT 18 0 "delta[0]" "6# &%&deltaG6#\"\"!" }{TEXT -1 1 "," }{XPPEDIT 18 0 "delta[1]" "6#&%&d {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 41 "recpoly, rechyper - Petkovsek's algorithm" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 88 "\n rec2poly(re,s(n)); \n recpoly(re,s(n));\n rec2hyper(re,s(n));\n rechyper(re,s(n)); " }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " } {TEXT 23 5 "re - " }{TEXT -1 83 "an equation or algebraic expression, \+ the recurrence equation in s(n),s(n+1),...\n " }{TEXT 23 4 "n - " } {TEXT -1 35 "a name, the recurrence variable\n " }{TEXT 23 4 "s - " }{TEXT -1 31 "a name, the recurrence function" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 222 "These functions are implementations of Petkovsek's algorithm. The procedure recpoly/rechyper searches for ClosedformFPSFPS,FPSFPS,FormalPowerSeriesFPS,HolonomicDE FPS,SimpleDE FPS,SimpleREFPS,standardsumFPSBriefdescriptionFormalPowerSeriesGFdiffeq GFrecursion HolonomicDESimpleDESimpleRE StandardSum Sumtohyper WZcertificate closedform contgosperextended_gosper fasenmyergosperhsumhsum,references hsum,theory hyperterm intdiffeq intrecursionkfreerecqqGAMMA qbinomial qbracketsqfac qfactorial qfunctionsqgosper qhypertermrealrec32    rechypreco recommend+ recordrecpo recrangrecurecurrecurr recurrenco     recurs+ recursi recursivereduc referreferenc'E "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Function:" }{TEXT 257 25 " qhyperterm - Produces a " }{XPPEDIT 18 0 "q" "6#%\"qG" }{T{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 on((-1)^k*binomial(n,k)*binomial(4*k,n),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#reG/,**,\"\"$\"\"\",&*&F(F)%\"nGF)F)\"\"(F)F),&*&F(F )F,F)F)\"\"%F)F),&*&F(F)F,F)F)\"\")F)F)-%\"sG6#,&F,F)F(F)F)F)**F0F)F.F ),(*&\"#PF))F,\"\"#F)F)*&\"$!=F)F,F)F)\"$=#F)F)-F56#,&F,F)F=F)F)F)**\" #;F)FCF),(*&\"#LF)F " 0 "" {MPLTEXT 1 0 18 "rechyper(re,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<#!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "rechyper((n+4)*s (n+2)+s(n+1)-(n+1)*s(n),s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,$ **,&*&\"\"#\"\"\"%\"nGF)F)\"\"&F)F),&F*F)F)F)F),&*&F(F)F*F)F)\"\"$F)! \"\",&F*F)F/F)F0F0*&F,F)F1F0" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "kfreerec and fasenmyer" 2 "fasenmyer" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "closedform and sumrecursion" 2 "sumrecursion " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "WZcere the WZ-method failed (see " } {HYPERLNK 17 "WZcertificate" 2 "WZcertificate" "" }{TEXT -1 49 ") and \+ sumrecursion returns a recurrence of order " }{XPPEDIT 18 0 "d" "6#%\" dG" }{TEXT -1 26 " that is not minimal (for " }{XPPEDIT 18 0 "d>1" "6# 2\"\"\"%\"dG" }{TEXT -1 4 ") as" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "su m((-1)^k*binomial(n,k)*binomial(d*k,n),k=0..n)=(-d)^n" "6#/-%$sumG6$*( ),$\"\"\"!\"\"%\"kGF*-%)binomialG6$%\"nGF,F*-F.6$*&%\"dGF*F,F*F0F*/F,; \"\"!F0),$F4F+F0" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "for a ll positive integers d." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "re:= sum recursion((-1)^k*binomial(n,k)*binomial(3*k,n),k,s(n));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#reG/,(*(\"\"#\"\"\",&*&F(F)%\"nGF)F)\"\"$F)F) -%\"sG6#,&F,F)F(F)F)F)*(F-F),&*&\"\"&F)F,F)F)\"\"(F)F)-F/6#,&F,F)F)F)F )F)*(\"\"*F)F9F)-F/6#F,F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rec2hyper(re,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#<#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "re:= sumrecursi elt aG6#\"\"\"" }{TEXT -1 90 ",... they can be considered as arbitrary con stants. (Note that those variables are local!)" }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfram~Koepf,~Univer sity~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "rec2poly (n*(n+1)*s(n+2)-2*n*(n+10)*s(n+1)+(n+9)*(n+10)*s(n),s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*6%\"nG\"\"\",&F$F%\"\")F%F%,&F$F%\"\"(F%F%, &F$F%\"\"'F%F%,&F$F%\"\"&F%F%,&F$F%\"\"%F%F%,&F$F%\"\"$F%F%,&F$F%\"\"# F%F%,&F$F%F%F%F%,(*&F$F%&%&deltaG6#\"\"!F%F%&F86#F%F%*&\"#OF%F7F%!\"\" F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "rec2poly(n*s(n+2)-2*n *s(n+1)+(n+10)*s(n),s(n));" }}{PARA 8 "" 1 "" {TEXT -1 51 "Error, (in \+ rec2poly) No polynomial solution exists\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Here an example whe polynomial/hypergeometric ter m solutions of a given homogeneous recurrence equation with polynomial coefficients in " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 135 ". The c orresponding procedures rec2poly and rec2hyper are special versions th at can only be used for recurrence equations of order two." }}{PARA 15 "" 0 "" {TEXT -1 125 "The output of the procedures is a set contain ing all possible solutions. I.e. if no solution exists an empty set is returned." }}{PARA 15 "" 0 "" {TEXT -1 13 "Note that if " }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#%\"nG" }{TEXT -1 94 " is a solution of the recu rrence equation the procedures rec2hyper and rechyper do NOT return " }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#%\"nG" }{TEXT -1 31 " but rather the \+ rational ratio " }{XPPEDIT 18 0 "f(n+1)/f(n)" "6#*&-%\"fG6#,&%\"nG\"\" \"F)F)F)-F%6#F(!\"\"" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 39 " If the solution contains any variables " }{XPPEDIT 18 0 "delta[0]" "6# &%&deltaG6#\"\"!" }{TEXT -1 1 "," }{XPPEDIT 18 0 "delta[1]" "6#&%&d {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 41 "recpoly, rechyper - Petkovsek's algorithm" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 88 "\n rec2poly(re,s(n)); \n recpoly(re,s(n));\n rec2hyper(re,s(n));\n rechyper(re,s(n)); " }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " } {TEXT 23 5 "re - " }{TEXT -1 83 "an equation or algebraic expression, \+ the recurrence equation in s(n),s(n+1),...\n " }{TEXT 23 4 "n - " } {TEXT -1 35 "a name, the recurrence variable\n " }{TEXT 23 4 "s - " }{TEXT -1 31 "a name, the recurrence function" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 222 "These functions are implementations of Petkovsek's algorithm. The procedure recpoly/rechyper searches fordependdepth derivat descridescrib descripdescriptBdesidesigndetdetectdeterminW,detlef;dev developm dformdg dgf'  0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 14 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT 257 2 " \+ " }{TEXT -1 30 "qsimpcomb - Simplification of " }{XPPEDIT 18 0 "q" "6 #%\"qG" }{TEXT -1 71 "-hypergeometric td{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 "sumdel tanabla(hahnterm,k,S(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*(,(%% betaG\"\"\"F(F(%\"xGF(F(,(F(F(%\"NG!\"\"F)F(F(-%&DeltaG6$-%&NablaG6$-% \"SG6#F)F)F)F(F,*&,0*&F)F(%&alphaGF(F(*&F)F(F'F(F(*&\"\"#F(F)F(F(F'F(F (F(*&F+F(F'F(F,F+F,F(F0F(F,*(F3F(%\"nGF(,*F9F(F'F(F?F(F(F(F(F(\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "`sumdelta+nabla`(hahnterm, k,S(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(,(%%betaG\"\"\"F(F(% \"xGF(F(,(F(F(%\"NG!\"\"F)F(F(-%&DeltaG6$-%\"SG6#F)F)F(F,*(,(F)F(%&alp haGF,F+F,F(F)F(-%&NablaGF/F(F(*(F0F(%\"nGF(,*F5F(F'F(F9F(F(F(F(F(\"\"! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Wilson:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "wilsonterm:=pochhammer(a+b,n)*pochhammer(a+c,n)*\npo chhammer(a+d,n)*\nhyperterm([-n,a+b+c+d+n-1,a-x,a+x],[a+b,a+c,a+d],1,k ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sumdeltanabla(wilsont erm,k,S(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*.,&*&\"\"#\"\"\"% \"xGF)F)F)!\"\"F),&F*F)%\"cGF)F),&%\"dGF)F*F)F),&%\"bGF)F*F)F),&%\"aGF )F*F)F)-%&DeltaG6$-%"" {TEXT -1 82 "Now the program can shift by the i nteger n, and we get the correct representation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Sumtohyper(binomial(2*n,n-k),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*HypergeomG6%7#,$*&\"\"#\"\"\"%#n|irGF*!\"\"7 \"F," }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " } {HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "hypert erm" 2 "hyperterm" "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } F ,F-F,F//F/;\"\"!*&F+F,F-F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 141 "where the first version without any assumptions results in a r epresentation with wrong bounds of the left-hand sum since it is not b ilateral." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Sumtohyper(binomial(2* n,n-k),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%)binomialG6$,$*&\"\" #\"\"\"%\"nGF*F*F+F*-%*HypergeomG6%7$,$F+!\"\"F*7#,&F+F*F*F*F1F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assume(n,integer);" }}} {EXCHG {PARA 0 "" 0 ,k)^2,k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&\"\"#\"\"\"%\"nGF'F'F'!\"\"F'-%* HypergeomG6%7%,&*&F&F'F(F'F)F&F',&F'F'F(F)F07$,&*&F&F'F(F'F)F'F'F&F'F' " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Here is an example for the us e of " }{HYPERLNK 17 "assume" 2 "assume" "" }{TEXT -1 1 ":" }{TEXT -1 17 " Consider the sum" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sum(binomial (2*n,n-k),k=-n..n)=sum(binomial(2*n,k),k=0..2*n)" "6#/-%$sumG6$-%)bino mialG6$*&\"\"#\"\"\"%\"nGF,,&F-F,%\"kG!\"\"/F/;,$F-F0F--F%6$-F(6$*&F+F ,F-F,F//F/;\"\"!*&F+F,F-F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 141 "where the first version without any assumptions results in a r epresentation with wrong bounds of the left-hand sum since it is not b ilateral." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Sumtohyper(binomial(2* n,n-k),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%)binomialG6$,$*&\"\" #\"\"\"%\"nGF*F*F+F*-%*HypergeomG6%7$,$F+!\"\"F*7#,&F+F*F*F*F1F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assume(n,integer);" }}} {EXCHG {PARA 0 "" 0 hyper returns the inert function Hypergeom (see also " } {HYPERLNK 17 "hypergeom" 2 "hypergeom" "" }{TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXT -1 110 "This procedure also uses some assumptions on the given indeterminates of the summand f that are specified via " } {HYPERLNK 17 "assume" 2 "assume" "" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfram~Koepf,~Univer sity~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Sumtohyp er(binomial(n,k),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*HypergeomG6 %7#,$%\"nG!\"\"7\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Sum tohyper(binomial(n,k)^2,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*Hype rgeomG6%7$,$%\"nG!\"\"F'7#\"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Sumtohyper(binomial(n,k)^2-binomial(n-10 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 65 "Sumtohyper - Express an infinite sum as a hypergeometric function" }} {PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 21 "\n \+ Sumtohyper(f, k);" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" } {TEXT -1 4 "\n " }{TEXT 23 4 "f - " }{TEXT -1 17 "an expression\n \+ " }{TEXT 23 4 "k - " }{TEXT -1 30 "a name, the summation variable" }}} {SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 40 "The function Sumtohyper converts the sum" }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "sum(f(k),k=-infinity..infinity)" "6#- %$sumG6$-%\"fG6#%\"kG/F);,$%)infinityG!\"\"F-" }{TEXT -1 1 "," }} {PARA 14 "" 0 "" {TEXT -1 170 "involving exponentials, factorials, bin omial coefficients, and Pochhammer symbols into hypergeometric notatio n. Sumtor{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 25 "Courier" 0 1 0 0 0 1 2 2 0 0 0 0 0 0 0 1 }{CSTYLE "Help Norma l" -1 30 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 " times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Head ing 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0Vnteger value (symbolic). The function is defined as: " }} {PARA 257 "" 0 "" {XPPEDIT 18 0 "qpochhammer(a,q,infinity)=product(1-a *q^j,j=0..infinity)" "6#/-%,qpochhammerG6%%\"aG%\"qG%)infinityG-%(prod uctG6$,&\"\"\"F.*&F'F.)F(%\"jGF.!\"\"/F1;\"\"!F)" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 2 "6#/-%,qpochhamm erG6%%\"aG%\"qG%\"kG-%*PIECEWISEG6%7$-%(productG6$,&\"\"\"F2*&F'F2)F(% \"jGF2!\"\"/F5;\"\"!,&F)F2F6F22F9F)7$F2/F)F97$-F/6$*$F1F6/F5;F)F22F)F9 " }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 5 "qfac:" }{TEXT -1 81 " This f unction is a synonym for qpochhammer, i.e. qfac(a,q,k)=qpochhammer(a,q ,k)." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "qbrackets:" }{TEXT -1 105 " The qbrackets function has two arguments. Both can be any valid \+ algebraic expression. The definition is:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "qbrackets(k,q) = (q^k-1)/(q-1)" "6#/-%*qbracketsG6$%\"k G%\"qG*&,&)F(F'\"\"\"F,!\"\"F,,&F(F,F,F-F-" }}}{SECT 0 {PARA 0 "" 0 " " {TEXT 26 10 "qbinomial:" }{TEXT -1 233 " The qbi~d{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0\6#,&%\"xG\" \"\"F,F,F,-%$tauG6#,&F+F,F,F,F,F,-%&DeltaG6$-&%\"pG6#%\"nG6#F+F+F,F,*& -F(6#F+F,-%&NablaG6$-&F66#F86#F+F+F,!\"\"*&&%'lambdaG6#F8F,-&F66#F86#F +F,F,\"\"!" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "Nabla(f( x),x)" "6#-%&NablaG6$-%\"fG6#%\"xGF)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Delta(f(x),x)" "6#-%&DeltaG6$-%\"fG6#%\"xGF)" }{TEXT -1 15 " are defined as" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "Nabla(f(x),x)=f(x)-f(x -1)" "6#/-%&NablaG6$-%\"fG6#%\"xGF*,&-F(6#F*\"\"\"-F(6#,&F*F.F.!\"\"F2 " }{TEXT -1 12 " and " }{XPPEDIT 18 0 "Delta(f(x),x)=f(x+1)-f(x )" "6#/-%&DeltaG6$-%\"fG6#%\"xGF*,&-F(6#,&F*\"\"\"F/F/F/-F(6#F*!\"\"" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 81 "As these functions are variations of Zeilberger's algorithm it is necessary that " } {XPPEDIT 18 0 "f(x,k)" "6#-%\"fG6$%\"xG%\"kG" }{TEXT -1 24 " is hyperg eometric wrt. " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 6 ", i.e." }}{PARA 14 "" 0 "" {XPPEDIT 18 0 "f(x+1,k)/f(x,k)tence variable\n " }{TEXT 23 4 "p - " }{TEXT -1 24 "a name, the sum function" }}}{SECT 0 {PARA 0 "" 0 "synop sis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 80 "These \+ functions are variations of implementations of Zeilberger's algorithm. For" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "p[n](x)=sum(f(x,k),k=-infinit y..infinity)" "6#/-&%\"pG6#%\"nG6#%\"xG-%$sumG6$-%\"fG6$F*%\"kG/F1;,$% )infinityG!\"\"F5" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 60 "sum deltanabla searches for a difference equation of the form" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sigma(x)*Delta(Nabla(p[n](x),x),x)+tau(x)*D elta(p[n](x),x)+lambda[n]*p[n](x)=0" "6#/,(*&-%&sigmaG6#%\"xG\"\"\"-%& DeltaG6$-%&NablaG6$-&%\"pG6#%\"nG6#F)F)F)F*F**&-%$tauG6#F)F*-F,6$-&F36 #F56#F)F)F*F**&&%'lambdaG6#F5F*-&F36#F56#F)F*F*\"\"!" }}{PARA 14 "" 0 "" {TEXT -1 40 "and `sumdelta+nabla` for one of the form" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "(sigma(x+1)+tau(x+1))*Delta(p[n](x),x)-sigma(x) *Nabla(p[n](x),x)+lambda[n]*p[n](x)=0" "6#/,(*&,&-%&sigmaG 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 14 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 72 "sumdeltanabla, `sumdelta+nabla` - a variation of Zeilberger's algori thm" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 61 "\n sumdeltanable(f, k, p(x));\n `sumdelta+nabla`(f,k,p(x));" } }{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 4 "f - " }{TEXT -1 17 "an expression\n " }{TEXT 23 4 "k - " } {TEXT -1 34 "a name, the summation variable\n " }{TEXT 23 4 "x - " } {TEXT -1 35 "a name, the differuibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctsumdeltanablasumdeltanablavariatzeilbergalgorithmusagcallsequencsumdeltanablparameterexpressnamesummatvariabldifferencsumsynopsisdescriptthesfunctionimplementatalgorithminfinitpgngxgsumgfgkginfinitygdeltanablasearchequatformsigmadeltataueltalambdasigmagdeltagnablagtauglambdagxgfdefinnecessarhypergeometricwrtkgfhaverationalsumdeltahomogeneoustatabovexamplreadhsummplvpackaghypergeometricmapleycopyrightwolframkoepfuniversitkasselghahnhahntermpochhammerpochhammbetanhypertermalphasumdeltanablabetagsgalphagfngfalphagfnablagfwilsonwilsontermnpochhammwilsontermcgfdgfbgfagffdfdffsftabgseealsoalsosumrecursmrecurshypertermintrecursintdiffeqsumdiffeqsumdiffrulsumintrulence variable\n " }{TEXT 23 4 "p - " }{TEXT -1 24 "a name, the sum function" }}}{SECT 0 {PARA 0 "" 0 "synop sis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 80 "These \+ functions are variations of implementations of Zeilberger's algorithm. For" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "p[n](x)=sum(f(x,k),k=-infinit y..infinity)" "6#/-&%\"pG6#%\"nG6#%\"xG-%$sumG6$-%\"fG6$F*%\"kG/F1;,$% )infinityG!\"\"F5" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 60 "sum deltanabla searches for a difference equation of the form" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sigma(x)*Delta(Nabla(p[n](x),x),x)+tau(x)*D elta(p[n](x),x)+lambda[n]*p[n](x)=0" "6#/,(*&-%&sigmaG6#%\"xG\"\"\"-%& DeltaG6$-%&NablaG6$-&%\"pG6#%\"nG6#F)F)F)F*F**&-%$tauG6#F)F*-F,6$-&F36 #F56#F)F)F*F**&&%'lambdaG6#F5F*-&F36#F56#F)F*F*\"\"!" }}{PARA 14 "" 0 "" {TEXT -1 40 "and `sumdelta+nabla` for one of the form" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "(sigma(x+1)+tau(x+1))*Delta(p[n](x),x)-sigma(x) *Nabla(p[n](x),x)+lambda[n]*p[n](x)=0" "6#/,(*&,&-%&sigmaGons3 ontopoperaoperator  operatorg optio option# optionaoptionalS'ord ordeordero9ordin orrespond orthogonal oseosedformosper ototherotherwiotherwisouqhypqhyperg  qhypergeomqhypergeometric qhypert qhypertermK,qlaguerr  qlaguerreg qp  qphihypertermqpoqpocqpochqpochhqpochha qpochham#rterm" 2 "hyperterm " "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intrecursion" 2 "intrecursion" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffeq" 2 "sumdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffrule and sumintrule" 2 "sumdiffrule" "" }{TEXT -1 1 " " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 } F)F/F)FSF)F )F+F)F:F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "`sumdel ta+nabla`(wilsonterm,k,S(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*. ,&*&\"\"#\"\"\"%\"xGF)F)F)!\"\"F),&F*F)%\"cGF)F),&%\"dGF)F*F)F),&%\"bG F)F*F)F),&%\"aGF)F*F)F)-%&DeltaG6$-%\"SG6#F*F*F)F+*.,&*&F(F)F*F)F)F)F) F),&F*F)F-F+F),&F*F)F/F+F),&F*F)F1F+F),&F3F)F*F+F)-%&NablaGF6F)F+*0F(F )%\"nGF)F*F)F&F)F;F),.F3F)F1F)F-F)F/F)FDF)F)F+F)F7F)F)\"\"!" }}}} {SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumrecursion" 2 "su mrecursion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "hypeS!{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 " " 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }&NablaG6$-%\"SG6#F*F*F*F)F+*(F(F),B*(F/F)F-F))F*F( F)F+*(F-F)F1F)F@F)F+*(F/F)F1F)F@F)F+*(F(F)F/F))F*\"\"%F)F)*(F(F)F3F)FD F)F)*$FDF)F+*(F(F)F-F)FDF)F)*,F(F)F/F)F-F)F1F)F@F)F)*(F(F)F1F)FDF)F)*, F(F)F/F)F3F)F1F)F@F)F)*,F(F)F-F)F3F)F1F)F@F)F)*(F3F)F1F)F@F)F+*,F(F)F/ F)F-F)F3F)F@F)F)*(F3F)F-F)F@F)F+**F3F)F/F)F-F)F1F)F+*(F3F)F/F)F@F)F+F) F7F)F+*0F(F)%\"nGF)F*F)F&F),&*&F(F)F*F)F)F)F)F),.F3F)F1F)F-F)F/F)FSF)F )F+F)F:F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "`sumdel ta+nabla`(wilsonterm,k,S(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*. ,&*&\"\"#\"\"\"%\"xGF)F)F)!\"\"F),&F*F)%\"cGF)F),&%\"dGF)F*F)F),&%\"bG F)F*F)F),&%\"aGF)F*F)F)-%&DeltaG6$-%\"SG6#F*F*F)F+*.,&*&F(F)F*F)F)F)F) F),&F*F)F-F+F),&F*F)F/F+F),&F*F)F1F+F),&F3F)F*F+F)-%&NablaGF6F)F+*0F(F )%\"nGF)F*F)F&F)F;F),.F3F)F1F)F-F)F/F)FDF)F)F+F)F7F)F)\"\"!" }}}} {SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumrecursion" 2 "su mrecursion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "hype "sumdel tanabla(hahnterm,k,S(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*(,(%% betaG\"\"\"F(F(%\"xGF(F(,(F(F(%\"NG!\"\"F)F(F(-%&DeltaG6$-%&NablaG6$-% \"SG6#F)F)F)F(F,*&,0*&F)F(%&alphaGF(F(*&F)F(F'F(F(*&\"\"#F(F)F(F(F'F(F (F(*&F+F(F'F(F,F+F,F(F0F(F,*(F3F(%\"nGF(,*F9F(F'F(F?F(F(F(F(F(\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "`sumdelta+nabla`(hahnterm, k,S(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(,(%%betaG\"\"\"F(F(% \"xGF(F(,(F(F(%\"NG!\"\"F)F(F(-%&DeltaG6$-%\"SG6#F)F)F(F,*(,(F)F(%&alp haGF,F+F,F(F)F(-%&NablaGF/F(F(*(F0F(%\"nGF(,*F5F(F'F(F9F(F(F(F(F(\"\"! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Wilson:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "wilsonterm:=pochhammer(a+b,n)*pochhammer(a+c,n)*\npo chhammer(a+d,n)*\nhyperterm([-n,a+b+c+d+n-1,a-x,a+x],[a+b,a+c,a+d],1,k ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sumdeltanabla(wilsont erm,k,S(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*.,&*&\"\"#\"\"\"% \"xGF)F)F)!\"\"F),&F*F)%\"cGF)F),&%\"dGF)F*F)F),&%\"bGF)F*F)F),&%\"aGF )F*F)F)-%&DeltaG6$-%" "6#*&-%\"fG6$,&%\"xG\"\"\"F)F)%\"kGF)- F%6$F(F*!\"\"" }{TEXT -1 13 " and " }{XPPEDIT 18 0 "f(x,k+1)/f (x,k)" "6#*&-%\"fG6$%\"xG,&%\"kG\"\"\"F*F*F*-F%6$F'F)!\"\"" }}{PARA 14 "" 0 "" {TEXT -1 23 "have to be rational in " }{XPPEDIT 18 0 "x" "6 #%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 1 " ." }}{PARA 15 "" 0 "" {TEXT -1 101 "The output of sumdeltanabla or `su mdelta+nabla` is a homogeneous difference equation as stated above." } }}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Ma ple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfr am~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Hahn" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "hahnterm:=(-1)^n/n!*pochh ammer(N-n,n)*pochhammer(beta+1,n)*\nhyperterm([-n,alpha+beta+n+1,-x],[ beta+1,1-N],1,k):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 316#,&%\"xG\" \"\"F,F,F,-%$tauG6#,&F+F,F,F,F,F,-%&DeltaG6$-&%\"pG6#%\"nG6#F+F+F,F,*& -F(6#F+F,-%&NablaG6$-&F66#F86#F+F+F,!\"\"*&&%'lambdaG6#F8F,-&F66#F86#F +F,F,\"\"!" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "Nabla(f( x),x)" "6#-%&NablaG6$-%\"fG6#%\"xGF)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Delta(f(x),x)" "6#-%&DeltaG6$-%\"fG6#%\"xGF)" }{TEXT -1 15 " are defined as" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "Nabla(f(x),x)=f(x)-f(x -1)" "6#/-%&NablaG6$-%\"fG6#%\"xGF*,&-F(6#F*\"\"\"-F(6#,&F*F.F.!\"\"F2 " }{TEXT -1 12 " and " }{XPPEDIT 18 0 "Delta(f(x),x)=f(x+1)-f(x )" "6#/-%&DeltaG6$-%\"fG6#%\"xGF*,&-F(6#,&F*\"\"\"F/F/F/-F(6#F*!\"\"" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 81 "As these functions are variations of Zeilberger's algorithm it is necessary that " } {XPPEDIT 18 0 "f(x,k)" "6#-%\"fG6$%\"xG%\"kG" }{TEXT -1 24 " is hyperg eometric wrt. " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 6 ", i.e." }}{PARA 14 "" 0 "" {XPPEDIT 18 0 "f(x+1,k)/f(x,k)),j=0..J)=G( x,k+1)-G(x,k)" "6#/-%$sumG6$*&-&%&sigmaG6#%\"jG6#%\"xG\"\"\"-%%diffG6$ -%\"fG6$F.%\"kG-%\"$G6$F.F,F//F,;\"\"!%\"JG,&-%\"GG6$F.,&F6F/F/F/F/-F@ 6$F.F6!\"\"" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 29 "By summin g over all integers " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 47 " we o btain a differential equation for the sum " }{XPPEDIT 18 0 "s(x)" "6#- %\"sG6#%\"xG" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "sum(sigma[j](x)*diff( s(x),x$j),j=0..J)=0" "6#/-%$sumG6$*&-&%&sigmaG6#%\"jG6#%\"xG\"\"\"-%%d iffG6$-%\"sG6#F.-%\"$G6$F.F,F//F,;\"\"!%\"JGF;" }{TEXT -1 1 "," }} {PARA 14 "" 0 "" {TEXT -1 58 "provided that the limit on the right-han d side exists with" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "limit(G(x,k),k= infinity)=limit(G(x,k),k=-infinity)" "6#/-%&limitG6$-%\"GG6$%\"xG%\"kG /F+%)infinityG-F%6$-F(6$F*F+/F+,$F-!\"\"" }}{PARA 14 "" 0 "" {TEXT -1 3 "and" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "diff(s(x),x)=sum(diff(f(x,k ),x),k=-infinity..infinity)" "6#/-%%diffG6$-%\"sG6#%\"xGF*-%$sumG6$-F% 6$-%\"fG6$F*%\"kGv" and hypergeometric wrt. " }{TEXT 262 1 "k" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 7 "A term " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 11 " is called " }{TEXT 257 17 "hyperexponential " }{TEXT -1 2 "or" }{TEXT 258 22 " hypergeometric wrt. x" }{TEXT -1 14 " if the ratio " }{XPPEDIT 18 0 "diff(g(x),x)" " 6#-%%diffG6$-%\"gG6#%\"xGF)" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "g(x)" "6# -%\"gG6#%\"xG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "g(x+1)" "6#-%\"gG6#, &%\"xG\"\"\"F(F(" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\" xG" }{TEXT -1 31 ", respectively, is rational in " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 59 "The algorith m is an application of Gosper's algorithm (see " }{HYPERLNK 17 "gosper " 2 "gosper" "" }{TEXT -1 10 " and also " }{HYPERLNK 17 "sumrecursion " 2 "sumrecursion" "" }{TEXT -1 94 ") thus calculating in a first step a differential equation for the summand f(x,k) of the form:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sum(sigma[j](x)*diff(f(x,k),x$jntial \+ equations" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" } {TEXT -1 24 "\n sumdiffeq(f,k,s(x));" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 4 "f - " }{TEXT -1 17 "an \+ expression\n " }{TEXT 23 4 "k - " }{TEXT -1 34 "a name, the summatio n variable\n " }{TEXT 23 4 "x - " }{TEXT -1 37 "a name, the differen tial variable\n " }{TEXT 23 4 "s - " }{TEXT -1 24 "a name, the sum f unction" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description: " }}{PARA 15 "" 0 "" {TEXT -1 145 "This function is an implementation \+ of a variation of Zeilberger's algorithm that tries to determine an or dinary differential equation for the sum" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "s(x)=sum(f(x,k),k=-infinity..infinity)" "6#/-%\"sG6#%\" xG-%$sumG6$-%\"fG6$F'%\"kG/F.;,$%)infinityG!\"\"F2" }{TEXT -1 1 "," }} {PARA 14 "" 0 "" {TEXT -1 14 "assuming that " }{XPPEDIT 18 0 "f(x,k)" "6#-%\"fG6$%\"xG%\"kG" }{TEXT -1 26 " is hyperexponential wrt. " } {TEXT 261 1 "x" }{TEXT -1 25 {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 259 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 78 "sumdiffeq - variation of Zeilberger's algorithm to find differeibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctsumdiffrulsumintrulvariatzeilbergalgorithmfindderivatintegratruleusagcallingsequencsumintparameterexpressnamesummatvariabldifferentialrecurrencsynopsidescriptletsuminfinitsgngxgsumgfgkgfinfinitygprocedurtriedeterminformdiffsigmadiffgxgfsigmagjgglobalmaxordgivemaximumshiftjgfcanappearrighthandsidedefaultsetbutyouassignanypositintegrocedurintegralintgigmagexamplreadhsummplvpackaghypergeometricycopyrightwolframkoepfuniversitkasselglegendrpolynomialbinomialbinomialpgbinomialgkginfinitygfbinomialntgseealalsocontgospintrecursintdiffeqgospsumdiffeqsumdeltanabladeltanablasumrecurszeilbrger#{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTY(%\"cGF(F(-%\"SG6#F&F(!\"\"**-F-6#F'F(%\"xGF(,&%\" bGF(F'F(F(,&%\"aGF(F'F(F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "assume(d>0,c>0,-1%#S0G-%%BetaG6$%#c|irG %#d|irG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "convert(S0,GAMMA );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%&GAMMAG6#%#c|irG\"\"\"-F%6#% #d|irGF(-F%6#,&F'F(F+F(!\"\"" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "contgosper" 2 "contgosper" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "rodriguesrecursion and rodriguesdiffeq" 2 "rodriguesrecursion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffrule and sumintrule" 2 "sumdiff rule" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumrecursion and zeilberger" 2 "sumrecursion" "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } "kG\"\" \"F(F(F(,(F'F(%\"dGFF+6#F'F(F'F(F(\" \"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Bateman Integral Represent ation: If " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 1 ">" }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT -1 1 ">" }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 1 "<" }{XPPEDIT 18 0 "x" "6#%\"x G" }{TEXT -1 1 "<" }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 6 " then\n " }{XPPEDIT 18 0 "int(t^(c-1)*(1-t)^(d-1)*hypergeom([a,b],[c],t*x),t=0 ..1)=GAMMA(c)*GAMMA(d)/GAMMA(c+d)" "6#/-%$intG6$*()%\"tG,&%\"cG\"\"\"F ,!\"\"F,),&F,F,F)F-,&%\"dGF,F,F-F,-%*hypergeomG6%7$%\"aG%\"bG7#F+*&F)F ,%\"xGF,F,/F);\"\"!F,*(-%&GAMMAG6#F+F,-F@6#F1F,-F@6#,&F+F,F1F,F-" } {TEXT -1 1 " " }{XPPEDIT 18 0 "hypergeom([a,b],[c+d],x)" "6#-%*hyperge omG6%7$%\"aG%\"bG7#,&%\"cG\"\"\"%\"dGF,%\"xG" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "intrecursion(t^(c-1)*(1-t)^(d-1)*\nhyperterm([a,b],[c ],t*x,k),t,S(k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(,&%\"kG\"\" \"F(F(F(,(F'F(%\"dGF 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric ~Summation\",~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Y Copyright~1998-2002,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "GAMMA" "6#%&GAMMAG" }{TEXT -1 14 "-func tion: " }{XPPEDIT 18 0 "GAMMA(z)=int(exp(-t)*t^(z-1),t=0..infinity) " "6#/-%&GAMMAG6#%\"zG-%$intG6$*&-%$expG6#,$%\"tG!\"\"\"\"\")F0,&F'F2F 2F1F2/F0;\"\"!%)infinityG" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "intrecursion(exp(-t)*t^(z-1),t,S(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%\"SG6#%\"zG\"\"\"F)F*F*-F'6#,&F)F*F*F*!\"\" \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Beta-function: " } {XPPEDIT 18 0 "B(z,w)=int(t^(z-1)*(1-t)^(w-1),t=0..1)" "6#/-%\"BG6$%\" zG%\"wG-%$intG6$*&)%\"tG,&F'\"\"\"F0!\"\"F0),&F0F0F.F1,&F(F0F0F1F0/F.; \"\"!F0" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "intrecur sion(t^(z-1)*(1-t)^(w-1),t,S(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,&*&,&%\"zG\"\"\"%\"wGF(F(-%\"SG6#,&F'F(F(F(F(!\"\"*&- }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } EXT -1 46 "Integration rule for the Legendre Polynomials " } {XPPEDIT 18 0 "P[n](x)" "6#-&%\"PG6#%\"nG6#%\"xG" }{TEXT -1 1 ":" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "sumintrule(binomial(n,k)*binomial(- n-1,k)*((1-x)/2)^k,k,P(n,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$I ntG6$-%\"PG6$%\"nG%\"xGF+,&*&,&*&\"\"#\"\"\"F*F1F1F1F1!\"\"-F(6$,&F1F2 F*F1F+F1F2*&F.F2-F(6$,&F*F1F1F1F+F1F1" }}}}{SECT 0 {PARA 0 "" 0 "seeal so" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "contgosper" 2 "contgosper" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "intrecursion" 2 "intrecursion" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "gosper" 2 "gosper" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffeq" 2 "sumdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdeltanabla" 2 "sum deltanabla" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumrecursion and zeilbe rger" 2 "sumrecursion" "" }}}}{MARK "0 0 0" 0+F.F1F1**F/F.-F(6$F/F+F.F0F1F2F1F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Integration rule for the Legendre Polynomials " } {XPPEDIT 18 0 "P[n](x)" "6#-&%\"PG6#%\"nG6#%\"xG" }{TEXT -1 1 ":" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "sumintrule(binomial(n,k)*binomial(- n-1,k)*((1-x)/2)^k,k,P(n,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$I ntG6$-%\"PG6$%\"nG%\"xGF+,&*&,&*&\"\"#\"\"\"F*F1F1F1F1!\"\"-F(6$,&F1F2 F*F1F+F1F2*&F.F2-F(6$,&F*F1F1F1F+F1F1" }}}}{SECT 0 {PARA 0 "" 0 "seeal so" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "contgosper" 2 "contgosper" "" }{TEXT -1 2 ", \+ " }{HYPERLNK 17 "intrecursion" 2 "intrecursion" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "gosper" 2 "gosper" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffeq" 2 "sumdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdeltanabla" 2 "sum deltanabla" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumrecursion and zeilbe rger" 2 "sumrecursion" "" }}}}{MARK "0 0 0" 0s(n,x),x)=sum(sigma[j](n )*s(n+j,x),j=-1..1)" "6#/-%$intG6$-%\"sG6$%\"nG%\"xGF+-%$sumG6$*&-&%&s igmaG6#%\"jG6#F*\"\"\"-F(6$,&F*F6F4F6F+F6/F4;,$F6!\"\"F6" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Mapl e~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-200 2,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Legendre Polynomials " }{XPPEDIT 18 0 "P[n](x)=sum(binom ial(n,k)*binomial(-n-1,k)*((1-x)/2)^k,k=-infinity..infinity)" "6#/-&% \"PG6#%\"nG6#%\"xG-%$sumG6$*(-%)binomialG6$F(%\"kG\"\"\"-F06$,&F(!\"\" F3F7F2F3)*&,&F3F3F*F7F3\"\"#F7F2F3/F2;,$%)infinityGF7F?" }{TEXT -1 1 " :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "sumdiffrule(binomial(n,k)*bino mial(-n-1,k)*((1-x)/2)^k,k,P(n,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%%diffG6$-%\"PG6$%\"nG%\"xGF+,&*,F+\"\"\",&F*F.F.F.F.F'F.,&F.!\"\"F +F.F1,&F.F.Fibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpeadingnormalheadlistitembulletfunctqrecsolvfindallhypergeometricsolutionrecurrencequatcallsequencqrecsolvreparameteralgebraicexpressquatnqnamenfnnvariablequationoptionalargumenttypedescriptprocedurdecideswhethgivenwithpolynomialcoefficientqgnganyhypergeometricfgnosolutioexistemptreturnotherwiswhereachsolutstoranothmayhomogeneouinhomogeneouoptiondeterminsortlookcansetpolynomialrationalqhypergeometricseridefaultprocduretrieformsuminfinitsumgagkgqgfinfinitygtermoptiowaydownratioupratiodownrecuprecqhypergeometricmeanupwardequatiofirstorderpossiblrecurrencdownratiospecifiedhypergeometricbutraththeirrationwhenusingsublistcontainentrelemsecondcontainrestrictglobalchangalsoqsumsettsplittrolsomeextentintroductadditionalnumberrequirsolvleadrailrecurshavefactorcompleteobtainfalsonlyfactorizationusunsparcsolarimaplinputcourimathtimehyperlinkoutputnormalheadbulletitempreolhelveticafontformalpowerserusagcallsequencformalpowerserexpreqndirvarordermethodfpsparametexpressequatnameoptionalirectleftrightrealcomplexarbutrecommendsummatvariablresultupperbounddesearchedhypergeometricrationalexplikinfodescriptfunctcanexpandmeromorphicfunctioncertaintypeintotheircorrespondlaurpuiseuxseriumtermforminfinitsymmetrnumbshiftointdevelopmfollowsupportnoeithhaveerivatsomewherintegfunctionsatisflinearhomogeneoudifferialwithconstantcoefficientformalpowersertriefindformalpowererieexpansrespectatpointalsoworkcaselogarithmicessentialsingularitfirstlookomogeneoudifferentialpolynomialhencderivatmustknownbelowyouexamplunidentificomputgivenargumcontroldepthsearchhighvaluwillincreaschancsolutcomplexitwelldefaultcomputeasymptoticmayaroundinfinityexperfesultinfinitys3) + .... which is correct in the context of asymptotic series. " }}{PARA 280 "" 0 "" {TEXT 285 173 "If dir is not specified, complex is chosen, except if the point of expansion is either infinity or -in finity. In this case real is taken as direction. See also limit[dirs]. " }}{PARA 281 "" 0 "" {TEXT 286 144 "If a method parameter is specifi ed, then the named method will be used. Currently accepted names are ' hypergeometric', 'explike' and 'rational'." }}{PARA 282 "" 0 "" {TEXT 287 69 "If eqn evaluates to a variable x then the equation x = 0 is as sumed. " }}{PARA 283 "" 0 "" {TEXT 288 55 "For a complete list of know n functions, see ?inifcns . " }}{PARA 284 "" 0 "" {TEXT 289 12 "REFERE NCES: " }}{PARA 285 "" 0 "" {TEXT 290 100 "Wolfram Koepf: Power Series in Computer Algebra, Journal of Symbolic Computation 13, 1992, 581-60 3. " }}{PARA 286 "" 0 "" {TEXT 291 190 "Wolfram Koepf: Algorithmic dev elopment of power series, Proceedings of the \"Conference on artificia l intelligence and symbolic co fastfastestfbfbffc# fcf#fdfdf fefferencfffff fg        fgffhfihowhowevhsuhsum    httphyphypehyper+hyperexponentialhyperg/hypergeo# hypergeom3 hypergeomet hypergeometr hypergeometri hypergeometric  1 34 "a recurrence equation for the sum " }{XPPEDIT 18 0 "s(n)" "6#-%\"sG6#%\"nG" }{TEXT -1 25 " on the left hand side..." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "zeilberger((-1)^k*binomial(n+b,n+k) *binomial(n+c,c+k)*binomial(b+c,b+k),k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,*%\"bG\"\"\"%\"nGF(F(F(%\"cGF(F(-%\"sG6#F)F(F(*&, &F(!\"\"F)F0F(-F,6#,&F)F(F(F(F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "...and the corresponding right hand side determined by cl osedform:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "closedform((-1)^k*bino mial(n+b,n+k)*binomial(n+c,c+k)*binomial(b+c,b+k),k,n);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*(-%)binomialG6$,&%\"bG\"\"\"%\"cGF)F(F)-%+pochh ammerG6$,(F(F)F*F)F)F)%\"nGF)-%*factorialG6#F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "An application of Closedform to deduce Clausen' s formula:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "Hypergeom([a,b],[a+b+1, 2],x)^2=Hypergeom([2*a,2*b,a+b],[a+b+1/2,2*a+2*b],x)" "6#/*$-%*Hyperge omG6%7$%\"aG%\"bG7$,(F)\"\"\"F*F-F-F-\"\"#%\"xGF.-FLs(n,x),x)=sum(sigma[j](n )*s(n+j,x),j=-1..1)" "6#/-%$intG6$-%\"sG6$%\"nG%\"xGF+-%$sumG6$*&-&%&s igmaG6#%\"jG6#F*\"\"\"-F(6$,&F*F6F4F6F+F6/F4;,$F6!\"\"F6" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Mapl e~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-200 2,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Legendre Polynomials " }{XPPEDIT 18 0 "P[n](x)=sum(binom ial(n,k)*binomial(-n-1,k)*((1-x)/2)^k,k=-infinity..infinity)" "6#/-&% \"PG6#%\"nG6#%\"xG-%$sumG6$*(-%)binomialG6$F(%\"kG\"\"\"-F06$,&F(!\"\" F3F7F2F3)*&,&F3F3F*F7F3\"\"#F7F2F3/F2;,$%)infinityGF7F?" }{TEXT -1 1 " :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "sumdiffrule(binomial(n,k)*bino mial(-n-1,k)*((1-x)/2)^k,k,P(n,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%%diffG6$-%\"PG6$%\"nG%\"xGF+,&*,F+\"\"\",&F*F.F.F.F.F'F.,&F.!\"\"F +F.F1,&F.F.F~XT -1 7 "Let " }{XPPEDIT 18 0 "s(n,x)=sum(f(n,k,x),x= -infinity..infinity)" "6#/-%\"sG6$%\"nG%\"xG-%$sumG6$-%\"fG6%F'%\"kGF( /F(;,$%)infinityG!\"\"F3" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 74 "The procedure sumdiffrule tries to determine a derivative rule of \+ the form" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(s(n,x),x)=sum(sigma[ j](n,x)*s(n+j,x),j=0..J)" "6#/-%%diffG6$-%\"sG6$%\"nG%\"xGF+-%$sumG6$* &-&%&sigmaG6#%\"jG6$F*F+\"\"\"-F(6$,&F*F6F4F6F+F6/F4;\"\"!%\"JG" } {TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 20 "The global variable " } {TEXT 256 8 "MAXORDER" }{TEXT -1 25 " gives the maximum shift " } {XPPEDIT 18 0 "s(n+j,x)" "6#-%\"sG6$,&%\"nG\"\"\"%\"jGF(%\"xG" }{TEXT -1 122 " that can appear on the right-hand side. By default it is set \+ to 5 but you can assign any positive integer or infinity to " }{TEXT 257 8 "MAXORDER" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 80 "The p rocedure sumintrule tries to determine an integration rule for the int egral" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "int(LE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 98 "sumdiffrule/sumintrule - variation of Zeilberger's algorithm to find \+ a derivative/integration rule" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Cal ling Sequence:" }{TEXT -1 55 "\n sumdiffrule(f,k,s(n,x));\n sumint rule(f,k,s(n,x));" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 4 "f - " }{TEXT -1 17 "an expression\n " } {TEXT 23 4 "k - " }{TEXT -1 34 "a name, the summation variable\n " } {TEXT 23 4 "x - " }{TEXT -1 37 "a name, the differential variable\n \+ " }{TEXT 23 4 "s - " }{TEXT -1 31 "a name, the recurrence function" }} }{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormaltextwarnerrorlistitemulletfunctcontgospcontinuouanalogugospalgorithmalmkvistzeilbergusagcallsequencparameterxpressnameintegrationvariablsynopsidescripimplementationversdueerhsumreferencreturnhyperexponentialantiderivativggxggivenfgwhenevsuchantiderivatexistdiffdiffgxgftermrationalthusproceduranalogonbuiltinintrestrictproblemfindexamplreamplvpackaghypergeometricsummatycopyrightwolframkoepfuniversitkasselgexpexpgcontgospnoseealsoalsointrecursintrcursintdiffeqsumdiffeqsumdiffrulsumintrul -1 7 "A term " }{XPPEDIT 18 0 "g(t)" "6#-%\"gG6#%\"tG" }{TEXT -1 11 " is called " }{TEXT 258 16 "h yperexponential" }{TEXT -1 14 " if the ratio " }{XPPEDIT 18 0 "diff(g( t),t)" "6#-%%diffG6$-%\"gG6#%\"tGF)" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "g (t)" "6#-%\"gG6#%\"tG" }{TEXT -1 16 " is rational in " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 40 "The algo rithm is an enhanced version of " }{HYPERLNK 17 "contgosper" 2 "contgo sper" "" }{TEXT -1 24 " applied to the function" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "F(t) = f(x,t)+Sum(sigma[j](x)*diff(f(x,t),`$`(x,j)),j = 1 .. j);" "6#/-%\"FG6#%\"tG,&-%\"fG6$%\"xGF'\"\"\"-%$SumG6$*&-&%&sigm aG6#%\"jG6#F,F--%%diffG6$-F*6$F,F'-%\"$G6$F,F6F-/F6;F-F6F-" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 19 "thus calculating a " }{TEXT 257 31 "hyperexponential antiderivative" }{TEXT -1 1 " " }{XPPEDIT 18 0 "G (x,t)" "6#-%\"GG6$%\"xG%\"tG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "F(t) " "6#-%\"FG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sigma[j](x)"mEXT -1 4 "\n " }{TEXT 23 4 "f - " }{TEXT -1 17 "an expression\n \+ " }{TEXT 23 4 "t - " }{TEXT -1 36 "a name, the integration variable\n \+ " }{TEXT 23 4 "x - " }{TEXT -1 37 "a name, the differential variable \n " }{TEXT 23 4 "s - " }{TEXT -1 32 "a name, the integration functi on" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }} {PARA 15 "" 0 "" {TEXT -1 150 "This function is an implementation of A lmkvist's and Zeilberger's algorithm that tries to determine a holonom ic differential equation for the integral" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "s(x)=int(f(x,t),x=a..b)" "6#/-%\"sG6#%\"xG-%$intG6$-%\" fG6$F'%\"tG/F';%\"aG%\"bG" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 14 "assuming that " }{XPPEDIT 18 0 "f(x,t)" "6#-%\"fG6$%\"xG%\"tG" }{TEXT -1 26 " is hyperexponential wrt. " }{XPPEDIT 18 0 "t" "6#%\"tG " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 24 " (co ntinuous version of " }{HYPERLNK 17 "sumdiffeq" 2 "sumdiffeq" "" } {TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXTibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctrodriguesrecursrodriguesdiffeqcalculatrecurrencdiffrentialequatgivenrodriguformulausagcallsequencrodriguesrecursparameterexpressnamesynopsidescriptthesfunctiondetermindifferentialfamilifgngxggivetypeggdiffghgutputdifferentialdiffeqapplicatintrecursintdiffeqrespectiveusingfollowintegralrepresentatpifactorialgpigfigfintgammaintgtggammagprocedurwilllookorderupmaxorderglobalvariablsetdefaultbutmaychanganypositintegvaluinfinitexamplreadhsummplvpackaghypergeometricsummatycopyrightwolframkoepfuniversitkasselgegendrpolynomialfactorialgnrecurrencsgxgfngfgenerallaguerrexpalphaexpgalphagfnrecurrencalphageealsoalsocontgospgfrecursgfdiffeqsumdiffrulsumintrulsumrecurszeilbergsumrecursion? {VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 " " 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 ence equatio n vanishes and returns the homogeneous version!" }}{PARA 15 "" 0 "" {TEXT -1 65 "The procedure will look for a recurrence equation of orde r up to " }{TEXT 256 8 "MAXORDER" }{TEXT -1 107 ". This global variabl e is set by default to 5 but may be changed to any positive integer va lue or infinity." }}{PARA 15 "" 0 "" {TEXT -1 29 "It is possible to ob tain the " }{TEXT 261 20 "rational certificate" }{TEXT -1 6 " (see " } {HYPERLNK 17 "hsum[theory]" 2 "hsum[theory]" "" }{TEXT -1 244 ") which allows a simple a posterio proof of the calculated recurrence equatio n. When using the optional argument certificate=true, intrecursion wil l return a list with two entries. The first is the computed recurrence equation for the integral " }{XPPEDIT 18 0 "s(n)" "6#-%\"sG6#%\"nG" } {TEXT -1 29 " and the second entry is the " }{TEXT 262 20 "rational ce rtificate" }{TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsu m6.mpl`;" }}{PARA"" 1 " " {XPPMATH 20 "6#,$*(,&*$)%\"xG\"\"#\"\"\"F*F*F*F*,**$)F(\"\"'F*F**$)F (\"\"&F*!\"\"F(F*F*F2F2,,*$)F(\"\"%F*F**$)F(\"\"$F*F*F&F*F(F*F*F*F2F2 " }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " } {HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "gosper " 2 "gosper" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intrecursion" 2 "intre cursion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intdiffeq" 2 "intdiffeq" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffeq" 2 "sumdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffrule and sumintrule" 2 "sumdiffrule" " " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } pG6#,$*&,&%\"xG\"\"\"F,F,F,,& F,!\"\"F+F,F.F.F,,&*$)F+\"\"#F,F,F,F,F,,&F,F.F0F,F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=diff((1+x^2)/(1-x^10),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*(\"\"#\"\"\"%\"xGF(,&F(F(*$)F )\"#5F(!\"\"F.F(**F-F(,&*$)F)F'F(F(F(F(F(F*!\"#F)\"\"*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "contgosper(f,x);" }}{PARA 11 #{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 " " 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }1 "" {XPPMATH 20 "6#*&%\"xG\"\"\"-%$expG6#,$*$)F $\"\"#F%!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "contgosp er(exp(-x^2)*(1-x^2),x);" }}{PARA 8 "" 1 "" {TEXT -1 65 "Error, (in co ntgosper) No hyperexponential antiderivative exists\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f:=diff(exp((1+x)/(1-x))*(1+x^2)/(1 -x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(**,&*&\"\"\"F),&F )F)%\"xG!\"\"F,F)*&,&F+F)F)F)F)F*!\"#F)F)-%$expG6#*&F.F)F*F,F),&*$)F+ \"\"#F)F)F)F)F),&F)F)F5F,F,F)**F7F)F0F)F+F)F8F,F)*,F7F)F0F)F4F)F8F/F+F )F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "contgosper(f,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(-%$expG6#,$*&,&%\"xG\"\"\"F,F,F,,& F,!\"\"F+F,F.F.F,,&*$)F+\"\"#F,F,F,F,F,,&F,F.F0F,F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=diff((1+x^2)/(1-x^10),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*(\"\"#\"\"\"%\"xGF(,&F(F(*$)F )\"#5F(!\"\"F.F(**F-F(,&*$)F)F'F(F(F(F(F(F*!\"#F)\"\"*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "contgosper(f,x);" }}{PARA 11 0 "" {TEXT -1 7 "and if " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 6 " i s a " }{TEXT 259 21 "hyperexponential term" }{TEXT -1 7 ", i.e. " } {XPPEDIT 18 0 "diff(g(x),x)" "6#-%%diffG6$-%\"gG6#%\"xGF)" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 16 " is ration al in " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 "." }}{PARA 15 "" 0 " " {TEXT -1 64 "Thus the procedure contgosper is an analogon to Maple's builtin " }{HYPERLNK 17 "int" 2 "int" "" }{TEXT -1 49 " procedure, re stricted to the problem of finding " }{TEXT 256 16 "hyperexponential" }{TEXT -1 17 " antiderivatives." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "rea d `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeo metric~Summation\",~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfram~Koepf,~University~of~KasselG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "contgosper(exp(-x^2)*(1-2*x^ 2),x);" }}{PARA 11 "" 6 "x - " }{TEXT -1 32 "a name, the integra tion variable" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Descrip tion:" }}{PARA 15 "" 0 "" {TEXT -1 88 "This function is an implementat ion of the continuous version of Gosper's algorithm (see " }{HYPERLNK 17 "gosper" 2 "gosper" "" }{TEXT -1 38 ") due to Almkvist and Zeilberg er (see " }{HYPERLNK 17 "hsum[references]" 2 "hsum[references]" "" } {TEXT -1 14 ") returning a " }{TEXT 258 31 "hyperexponential antideriv ative" }{TEXT -1 1 " " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 21 " of a given function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 40 " whenever such an antiderivative exists." }}{PARA 15 "" 0 "" {TEXT -1 11 "A function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG " }{TEXT -1 11 " is called " }{TEXT 257 31 "hyperexponential antideriv ative" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 3 " if" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(g(x),x)=f(x) " "6#/-%%diffG6$-%\"gG6#%\"xGF*-%\"fG6#F*" }}{PARA 14 "" maxmaximum maxo maxord/ may_!mdelta mdiffeqmemean mechanismmer# meromorphic mesmessagmetmethod/metric#mgseem selselgsent seq sequsequenc<seriCDsessset[G settsgs    shiftnthenthunticalntingntinuounull numb number nuppnwhennwhernxnzobobsoletobtain+ occuroccurrocedurochhamm inteintegVinteger?integraintegral integrat intel50 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 14 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 128 "intdiffeq - Algorithm of Almkvist and Zeilberger algorithm to det ermine a holonomic differential equation for definite integrals" }} {PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 24 "\n \+ intdiffeq(f,t,s(x));" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" } {T" }{XPPEDIT 18 0 "G(n,t)" "6#-%\"GG6$%\"nG% \"tG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(n,t)" "6#-%\"fG6$%\"nG%\"t G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sigma[j](n)" "6#-&%&sigmaG6#%\" jG6#%\"nG" }{TEXT -1 11 " such that:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "sum(sigma[j](n)*f(n+j,t),j=0..J)=diff(G(n,t),t)" "6#/-%$sumG6$*&-&% &sigmaG6#%\"jG6#%\"nG\"\"\"-%\"fG6$,&F.F/F,F/%\"tGF//F,;\"\"!%\"JG-%%d iffG6$-%\"GG6$F.F4F4" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 34 " By integrating this equation over " }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT -1 6 " from " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 42 " we get the resulting recurr ence equation:" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "sum(sigma[j](n)*s(n +j),j=0..J)=G(n,b)-G(n,a)" "6#/-%$sumG6$*&-&%&sigmaG6#%\"jG6#%\"nG\"\" \"-%\"sG6#,&F.F/F,F/F//F,;\"\"!%\"JG,&-%\"GG6$F.%\"bGF/-F:6$F.%\"aG!\" \"" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 134 "Warning: The algo rithm assumes that the right-hand side of the above recurrerge ometric" }{TEXT -1 6 " wrt. " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 24 " (continuous version of " }{HYPERLNK 17 "sumrecursion" 2 "sumrecur sion" "" }{TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXT -1 17 "Remember: A \+ term " }{XPPEDIT 18 0 "g(t)" "6#-%\"gG6#%\"tG" }{TEXT -1 11 " is calle d " }{TEXT 260 16 "hyperexponential" }{TEXT -1 14 " if the ratio " } {XPPEDIT 18 0 "diff(g(t),t)" "6#-%%diffG6$-%\"gG6#%\"tGF)" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "g(t)" "6#-%\"gG6#%\"tG" }{TEXT -1 16 " is ration al in " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 1 "." }}{PARA 15 "" 0 " " {TEXT -1 40 "The algorithm is an enhanced version of " }{HYPERLNK 17 "contgosper" 2 "contgosper" "" }{TEXT -1 24 " applied to the functi on" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "F(t)=sum(sigma[j](n)*f(n+j,t),j =0..J)" "6#/-%\"FG6#%\"tG-%$sumG6$*&-&%&sigmaG6#%\"jG6#%\"nG\"\"\"-%\" fG6$,&F2F3F0F3F'F3/F0;\"\"!%\"JG" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 19 "thus calculating a " }{TEXT 257 31 "hyperexponential anti derivative" }{TEXT -1 1 " ng Sequence:" }{TEXT -1 27 "\n intrecurs ion(f,t,s(n));" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 4 "f - " }{TEXT -1 17 "an expression\n " }{TEXT 23 4 "t - " }{TEXT -1 36 "a name, the integration variable\n " } {TEXT 23 4 "n - " }{TEXT -1 35 "a name, the recurrence variable\n " }{TEXT 23 4 "s - " }{TEXT -1 32 "a name, the integration function" }}} {SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 148 "This function is an implementation of Almkvist's and Zeilberger's algorithm that tries to determine a holonomic recurr ence equation for the integral" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "s(n )=int(f(n,t),t=-a..b)" "6#/-%\"sG6#%\"nG-%$intG6$-%\"fG6$F'%\"tG/F.;,$ %\"aG!\"\"%\"bG" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 14 "assum ing that " }{XPPEDIT 18 0 "f(n,t)" "6#-%\"fG6$%\"nG%\"tG" }{TEXT -1 4 " is " }{TEXT 258 16 "hyperexponential" }{TEXT -1 6 " wrt. " } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 5 " and " }{TEXT 259 14 "hyp {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 259 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 119 "intrecursion - Algorithm of Almkvist and Zeilberger to determine \+ a holonomic recurrence equation for definite integrals" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Callibinombinomibinomiabinomialc   binomialg[3bleboe boeingGbookbothbou17 "FPS[SimpleR E]" 2 "FPS[SimpleRE]" "" }{TEXT 297 2 ", " }{HYPERLNK 17 "FPS[standard sum]" 2 "standardsum" "" }{TEXT 305 2 ", " }{HYPERLNK 17 "inifcns" 2 " inifcns" "" }{TEXT 298 2 ", " }{HYPERLNK 17 "taylor" 2 "taylor" "" } {TEXT 299 2 ", " }{HYPERLNK 17 "series" 2 "series" "" }{TEXT 300 2 ", \+ " }{HYPERLNK 17 "Sum" 2 "Sum" "" }}}}{MARK "12 1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } > " 0 "" {MPLTEXT 1 0 93 "FPS((-1/2*x+1/6*x^3)*arcta n(x)+(-1/4*x^2+1/12)*\nln(x^2+1)+5/12*x^2+1/4, x=0, hypergeometric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"\"%F%*&F$F%-%$SumG6$*.)! \"\"%\"kGF%,&F.F%\"\"#F%F-,&*&F0F%F.F%F%F%F%F-,&F.F%F%F%F-,&\"\"$F%*&F 0F%F.F%F%F-)%\"xG,&*&F0F%F.F%F%F&F%F%/F.;\"\"!%)infinityGF%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 297 "" 0 "seealso" {TEXT 294 10 "See Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "FP S" 2 "FPS" "seealso" }{TEXT 295 2 ", " }{HYPERLNK 17 "FPS[HolonomicDE] " 2 "FPS[HolonomicDE]" "" }{TEXT 296 2 ", " }{HYPERLNK  14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 125 "rodriguesrecursion/rodriguesdiffeq - Calculate a recurrence/diffe rential equation for a function given by a Rodrigues formula" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 67 "\n rodr iguesrecursion(g,h,x,s(n));\n rodriguesdiffeq(g,h,n,s(x));" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 6 "g ,h - " }{TEXT -1 17 "an expression\n " }{TEXT 23 8 "s,x,n - " } {TEXT -1 6 "a name" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "De scription:" }}{PARA 15 "" 0 "" {TE 2 "sumrecursion " "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 } F'F&F'*&%\"aGF 'F&F'F'F'-F+6$F-F&F'F'*(F-F'F:F'F8F'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 80 "(See example Bateman Integral relation in Hypergeometri c Summation by W. Koepf.)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "intrec ursion(t^(c-1)*(1-t)^(d-1)*hyperterm([a,b],[c],t*x,k),t,S(k));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(,&%\"kG\"\"\"F(F(F(,(%\"cGF(F'F(% \"dGF(F(-%\"SG6#F&F(!\"\"**-F-6#F'F(%\"xGF(,&%\"bGF(F'F(F(,&%\"aGF(F'F (F(F(\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: \+ " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "con tgosper" 2 "contgosper" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intrecursio n" 2 "intrecursion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "rodriguesrecurs ion and rodriguesdiffeq" 2 "rodriguesrecursion" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "sumdiffrule and sumintrule" 2 "sumdiffrule" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumrecursion and zeilberger" ATH 20 "6#/,(*(%\"xG\"\"\",&F&F'F'!\"\"F'-%%diff G6$-%\"SG6#F&-%\"$G6$F&\"\"#F'F'*&,*%\"cGF)*&%\"bGF'F&F'F'F&F'*&%\"aGF 'F&F'F'F'-F+6$F-F&F'F'*(F-F'F:F'F8F'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 80 "(See example Bateman Integral relation in Hypergeometri c Summation by W. Koepf.)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "intrec ursion(t^(c-1)*(1-t)^(d-1)*hyperterm([a,b],[c],t*x,k),t,S(k));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(,&%\"kG\"\"\"F(F(F(,(%\"cGF(F'F(% \"dGF(F(-%\"SG6#F&F(!\"\"**-F-6#F'F(%\"xGF(,&%\"bGF(F'F(F(,&%\"aGF(F'F (F(F(\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: \+ " }{HYPERLNK 17 "hsum" 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "con tgosper" 2 "contgosper" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intrecursio n" 2 "intrecursion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "rodriguesrecurs ion and rodriguesdiffeq" 2 "rodriguesrecursion" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "sumdiffrule and sumintrule" 2 "sumdiffrule" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumrecursion and zeilberger" `;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\" ,~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~19 98-2002,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Consider: " }{XPPEDIT 18 0 "S(x)=int(x^2/((x^4+t^2) *(1+t^2)),x=-infinity..infinity)" "6#/-%\"SG6#%\"xG-%$intG6$*&F'\"\"#* &,&*$F'\"\"%\"\"\"*$%\"tGF,F1F1,&F1F1*$F3F,F1F1!\"\"/F';,$%)infinityGF 6F:" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "intdiffeq(x^ 2/((x^4+t^2)*(1+t^2)),t,S(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(* ,,&%\"xG\"\"\"F(!\"\"F(,&F'F(F(F(F(,&*$)F'\"\"#F(F(F(F(F(-%%diffG6$-% \"SG6#F'-%\"$G6$F'F.F(F'F(F(*&,&F(F(*&\"\"(F()F'\"\"%F(F(F(-F06$F2F'F( F(*(\"\")F(F2F()F'\"\"$F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "(See example Euler Integral relation in Hypergeometric Summation b y W. Koepf.)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "intdiffeq(GAMMA(c)/ (GAMMA(b)*GAMMA(c-b))*t^(b-1)*(1-t)^(c-b-1)*(1-t*x)^(-a),t,S(x));" }} {PARA 11 "" 1 "" {XPPMf the above recurrence equation vanishes \+ and returns the homogeneous version!" }}{PARA 15 "" 0 "" {TEXT -1 67 " The procedure will look for a differential equation of order up to " } {TEXT 256 8 "MAXORDER" }{TEXT -1 95 ". This global variable is set by \+ default to 5 but may be changed to any positive integer value." }} {PARA 15 "" 0 "" {TEXT -1 29 "It is possible to obtain the " }{TEXT 259 20 "rational certificate" }{TEXT -1 6 " (see " }{HYPERLNK 17 "hsum [theory]" 2 "hsum[theory]" "" }{TEXT -1 245 ") which allows a simple a posterio proof of the calculated differential equation. When using th e optional argument certificate=true, intdiffeq will return a list wit h two entries. The first is the computed differential equation for the integral " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 29 " and the second entry is the " }{TEXT 260 20 "rational certificate" } {TEXT -1 1 "." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpling an interface that permits it)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "read \"FPS.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %TCopyright~1995,~Dominik~Gruntz,~University~of~BaselG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%^oCopyright~2002,~Detlef~M|gzller~&~Wolfram~Koepf ,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 0 {PARA 4 "" 0 "seealso" {TEXT 263 10 "See Also: " }} {PARA 15 "" 0 "" {HYPERLNK 17 "FPS[FormalPowerSeries]" 2 "FPS[FormalPo werSeries]" "" }{TEXT 264 2 ", " }{HYPERLNK 17 "FPS[HolonomicDE]" 2 "F PS[HolonomicDE]" "" }{TEXT 265 2 ", " }{HYPERLNK 17 "FPS[SimpleRE]" 2 "FPS[SimpleRE]" "" }{TEXT 266 2 ", " }{HYPERLNK 17 "FPS[standardsum]" 2 "FPS[standardsum]" "" }{TEXT 268 2 ", " }{HYPERLNK 17 "with" 2 "with " "" }{TEXT 267 2 ". " }}}}{MARK "2 2 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } RA 15 "" 0 "" {TEXT 262 146 "To load this package, type the followi ng Maple commands in a Maple session (or cut and paste them, if you ar e usnomial function uses three arguments, where the first two are any valid algebraic expressi ons representing integer values (symbolic) and the last one is a name \+ or a name raised to an integer power. The function is defined as:" }} {PARA 259 "" 0 "" {XPPEDIT 18 0 "qbinomial(n,k,q) = qpochhammer(q,q,n) /(qpochhammer(q,q,k)*qpochhammer(q,q,n-k)) " "6#/-%*qbinomialG6%%\"nG% \"kG%\"qG*&-%,qpochhammerG6%F)F)F'\"\"\"*&-F,6%F)F)F(F.-F,6%F)F),&F'F. F(!\"\"F.F5" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "qfactorial:" } {TEXT -1 222 " The qfactorial function has two arguments. The first on e is any valid algebraic expression that represents an integer value ( symbolic) and the second one is a name or a name raised to any integer power. It is defined as: " }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "qfactor ial(k,q) = qpochhammer(q,q,k)/(1-q)^k" "6#/-%+qfactorialG6$%\"kG%\"qG* &-%,qpochhammerG6%F(F(F'\"\"\"),&F-F-F(!\"\"F'F0" }}}{SECT 0 {PARA 0 " " 0 "" {TEXT 26 7 "qGAMMA:" }{TEXT -1 108 " The qGAMMA function has tw o argumXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Generalized Laguerre Polynomials: " } {XPPEDIT 18 0 "f[n](x)=exp(x)/n!/x^alpha" "6#/-&%\"fG6#%\"nG6#%\"xG*(- %$expG6#F*\"\"\"-%*factorialG6#F(!\"\")F*%&alphaGF3" }{TEXT -1 1 " " } {XPPEDIT 18 0 "diff(exp(-x)*x^(alpha+n),x$n)" "6#-%%diffG6$*&-%$expG6# ,$%\"xG!\"\"\"\"\")F+,&%&alphaGF-%\"nGF-F--%\"$G6$F+F1" }{TEXT -1 39 " ,\nrecurrence and differential equation:" }{TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 67 "rodriguesrecursion(exp(x)/(n!*x^alpha),exp(-x) *x^(alpha+n),x,S(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG \"\"\"\"\"#F(F(-%\"SG6#F&F(F(*&,**&F)F(F'F(!\"\"%\"xGF(%&alphaGF0\"\"$ F0F(-F+6#,&F'F(F(F(F(F(*&,(F2F(F'F(F(F(F(-F+6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "rodriguesdiffeq(exp(x)/(n!*x^alpha) ,exp(-x)*x^(alpha+n),n,S(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*& %\"xG\"\"\"-%%diffG6$-%\"SG6#F&-%\"$G6$F&\"\"#F'F'*&,(F&F'%&alphaG!\" \"F'F5F'-F)6$F+F&F'F5*&F+F'%\"nGF'F'\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "s eealso" {TEXT 26 10 "Sometric~Summation\",~Maple~V~-~Maple ~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfram~ Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "L egendre Polynomials: " }{XPPEDIT 18 0 "f[n](x)=(-1)^n/(2^n*n!)" "6#/- &%\"fG6#%\"nG6#%\"xG*&),$\"\"\"!\"\"F(F.*&)\"\"#F(F.-%*factorialG6#F(F .F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "diff((1-x^2)^n,x$n)" "6#-%%diffG6 $),&\"\"\"F(*$%\"xG\"\"#!\"\"%\"nG-%\"$G6$F*F-" }{TEXT -1 39 ",\nrecur rence and differential equation:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "rodriguesrecursion((-1)^n/(2^n*n!),(1-x^2)^n,x,S(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\"#F(F(-%\"SG6#F&F(F(*(%\"xGF( ,&*&F)F(F'F(F(\"\"$F(F(-F+6#,&F'F(F(F(F(!\"\"*&F4F(-F+6#F'F(F(\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "rodriguesdiffeq((-1)^n/(2^ n*n!),(1-x^2)^n,n,S(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(,&\" \"\"!\"\"%\"xGF'F',&F'F'F)F'F'-%%diffG6$-%\"SG6#F)-%\"$G6$F)\"\"#F'F(* (F4F'F)F'-F,6$F.F)F'F(*(F.F'%\"nGF',&F9F'F'F'F'F'\"\"!" }}}{E " 2 "intrecursion" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 64 " respectively by using the following inte gral representation of " }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#% \"xG" }{TEXT -1 1 ":" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "f[n](x)=g[n]( x)*n!/(2*Pi*I)" "6#/-&%\"fG6#%\"nG6#%\"xG*(-&%\"gG6#F(6#F*\"\"\"-%*fac torialG6#F(F1*(\"\"#F1%#PiGF1%\"IGF1!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(h[n](t)/(t-x)^(n+1),t=gamma.._)" "6#-%$intG6$*&-&%\"hG6#%\"n G6#%\"tG\"\"\"),&F-F.%\"xG!\"\",&F+F.F.F.F2/F-;%&gammaG%\"_G" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 79 "The procedures will look for \+ a recurrence/differential equation of order up to " }{TEXT 256 8 "MAXO RDER" }{TEXT -1 107 ". This global variable is set by default to 5 but may be changed to any positive integer value or infinity." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hyperge XT -1 88 "These functions determine \+ a recurrence/differential equation for a familiy of functions " } {XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#%\"xG" }{TEXT -1 36 ", give n by a Rodrigues type formula:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f[n ](x)=g[n](x)*diff(h[n](x),x$n)" "6#/-&%\"fG6#%\"nG6#%\"xG*&-&%\"gG6#F( 6#F*\"\"\"-%%diffG6$-&%\"hG6#F(6#F*-%\"$G6$F*F(F1" }{TEXT -1 1 "," }} {PARA 14 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "g[n](x)" "6#-&%\" gG6#%\"nG6#%\"xG" }{TEXT -1 1 "=" }{TEXT 257 1 "g" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "h[n](x" "6#-&%\"hG6#%\"nG6#%\"xG" }{TEXT -1 2 ")=" } {TEXT 258 1 "h" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 104 "The o utput of rodriguesrecursion/rodriguesdiffeq is a recurrence/differenti al equation for the function " }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#% \"nG6#%\"xG" }{TEXT -1 1 "=" }{TEXT 259 14 "g*diff(h,x$n)." }}{PARA 15 "" 0 "" {TEXT -1 87 "The functions rodriguesrecursion and rodrigues diffeq are applications of the functions " }{HYPERLNK 17 "intrecursion ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctrodriguesrecursrodriguesdiffeqcalculatrecurrencdiffrentialequatgivenrodriguformulausagcallsequencrodriguesrecursparameterexpressnamesynopsidescriptthesfunctiondetermindifferentialfamilifgngxggivetypeggdiffghgutputdifferentialdiffeqapplicatintrecursintdiffeqrespectiveusingfollowintegralrepresentatpifactorialgpigfigfintgammaintgtggammagprocedurwilllookorderupmaxorderglobalvariablsetdefaultbutmaychanganypositintegvaluinfinitexamplreadhsummplvpackaghypergeometricsummatycopyrightwolframkoepfuniversitkasselgegendrpolynomialfactorialgnrecurrencsgxgfngfgenerallaguerrexpalphaexpgalphagfnrecurrencalphageealsoalsocontgospgfrecursgfdiffeqsumdiffrulsumintrulsumrecurszeilbergsumrecursion{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 XCHG {PARA 0 "" 0 "" {TEXT -1 37 "Generalized Laguerre Polynomials: " } {XPPEDIT 18 0 "f[n](x)=exp(x)/n!/x^alpha" "6#/-&%\"fG6#%\"nG6#%\"xG*(- %$expG6#F*\"\"\"-%*factorialG6#F(!\"\")F*%&alphaGF3" }{TEXT -1 1 " " } {XPPEDIT 18 0 "diff(exp(-x)*x^(alpha+n),x$n)" "6#-%%diffG6$*&-%$expG6# ,$%\"xG!\"\"\"\"\")F+,&%&alphaGF-%\"nGF-F--%\"$G6$F+F1" }{TEXT -1 39 " ,\nrecurrence and differential equation:" }{TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 67 "rodriguesrecursion(exp(x)/(n!*x^alpha),exp(-x) *x^(alpha+n),x,S(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG \"\"\"\"\"#F(F(-%\"SG6#F&F(F(*&,**&F)F(F'F(!\"\"%\"xGF(%&alphaGF0\"\"$ F0F(-F+6#,&F'F(F(F(F(F(*&,(F2F(F'F(F(F(F(-F+6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "rodriguesdiffeq(exp(x)/(n!*x^alpha) ,exp(-x)*x^(alpha+n),n,S(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*& %\"xG\"\"\"-%%diffG6$-%\"SG6#F&-%\"$G6$F&\"\"#F'F'*&,(F&F'%&alphaG!\" \"F'F5F'-F)6$F+F&F'F5*&F+F'%\"nGF'F'\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "s eealso" {TEXT 26 10 "S{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal " -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 ometric~Summation\",~Maple~V~-~Maple ~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfram~ Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "L egendre Polynomials: " }{XPPEDIT 18 0 "f[n](x)=(-1)^n/(2^n*n!)" "6#/- &%\"fG6#%\"nG6#%\"xG*&),$\"\"\"!\"\"F(F.*&)\"\"#F(F.-%*factorialG6#F(F .F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "diff((1-x^2)^n,x$n)" "6#-%%diffG6 $),&\"\"\"F(*$%\"xG\"\"#!\"\"%\"nG-%\"$G6$F*F-" }{TEXT -1 39 ",\nrecur rence and differential equation:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "rodriguesrecursion((-1)^n/(2^n*n!),(1-x^2)^n,x,S(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\"#F(F(-%\"SG6#F&F(F(*(%\"xGF( ,&*&F)F(F'F(F(\"\"$F(F(-F+6#,&F'F(F(F(F(!\"\"*&F4F(-F+6#F'F(F(\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "rodriguesdiffeq((-1)^n/(2^ n*n!),(1-x^2)^n,n,S(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(,&\" \"\"!\"\"%\"xGF'F',&F'F'F)F'F'-%%diffG6$-%\"SG6#F)-%\"$G6$F)\"\"#F'F(* (F4F'F)F'-F,6$F.F)F'F(*(F.F'%\"nGF',&F9F'F'F'F'F'\"\"!" }}}{E" 2 "intrecursion" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 64 " respectively by using the following inte gral representation of " }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#% \"xG" }{TEXT -1 1 ":" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "f[n](x)=g[n]( x)*n!/(2*Pi*I)" "6#/-&%\"fG6#%\"nG6#%\"xG*(-&%\"gG6#F(6#F*\"\"\"-%*fac torialG6#F(F1*(\"\"#F1%#PiGF1%\"IGF1!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(h[n](t)/(t-x)^(n+1),t=gamma.._)" "6#-%$intG6$*&-&%\"hG6#%\"n G6#%\"tG\"\"\"),&F-F.%\"xG!\"\",&F+F.F.F.F2/F-;%&gammaG%\"_G" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 79 "The procedures will look for \+ a recurrence/differential equation of order up to " }{TEXT 256 8 "MAXO RDER" }{TEXT -1 107 ". This global variable is set by default to 5 but may be changed to any positive integer value or infinity." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"HypergeXT -1 88 "These functions determine \+ a recurrence/differential equation for a familiy of functions " } {XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#%\"xG" }{TEXT -1 36 ", give n by a Rodrigues type formula:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f[n ](x)=g[n](x)*diff(h[n](x),x$n)" "6#/-&%\"fG6#%\"nG6#%\"xG*&-&%\"gG6#F( 6#F*\"\"\"-%%diffG6$-&%\"hG6#F(6#F*-%\"$G6$F*F(F1" }{TEXT -1 1 "," }} {PARA 14 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "g[n](x)" "6#-&%\" gG6#%\"nG6#%\"xG" }{TEXT -1 1 "=" }{TEXT 257 1 "g" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "h[n](x" "6#-&%\"hG6#%\"nG6#%\"xG" }{TEXT -1 2 ")=" } {TEXT 258 1 "h" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 104 "The o utput of rodriguesrecursion/rodriguesdiffeq is a recurrence/differenti al equation for the function " }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#% \"nG6#%\"xG" }{TEXT -1 1 "=" }{TEXT 259 14 "g*diff(h,x$n)." }}{PARA 15 "" 0 "" {TEXT -1 87 "The functions rodriguesrecursion and rodrigues diffeq are applications of the functions " }{HYPERLNK 17 "intrecursion  14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "" 14 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 125 "rodriguesrecursion/rodriguesdiffeq - Calculate a recurrence/diffe rential equation for a function given by a Rodrigues formula" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 67 "\n rodr iguesrecursion(g,h,x,s(n));\n rodriguesdiffeq(g,h,n,s(x));" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 6 "g ,h - " }{TEXT -1 17 "an expression\n " }{TEXT 23 8 "s,x,n - " } {TEXT -1 6 "a name" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "De scription:" }}{PARA 15 "" 0 "" {TEibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalbulletitemlistfunctgfrecursgfdiffeqcalculatrecurrencdifferentialequationgivengeneratusagecallsequencparameterexpressnamesynopsidescriptthesfunctiondetermindifferentialfamilifgngxgzgwithsuminfinitysumgaginfinitygprocedurapplicatheintrecursintdiffeqthushyperexponentialwrtwilllookorderupmaxordthglobalvariablsetdefaultbutmaychanganypositivintegvaluinfinitexamplreahsummplvpackaghypergeometricsummatycopyrightwolframkoepfuniversitkasselggenerallaguerrpolynomialwialphaexpzgfalphagfexpgexlaguerregalphagxgfalphadiffgngfseealsoalsocontgosprodriguesrecursrodriguesdiffeqrodrigusumdiffrulsumintrulsumrecurszeilberg 17 "qsum[t heory]" 2 "qsum[theory]" "" }{TEXT -1 23 "). The global variable " } {TEXT 19 17 "_qsum_solvemethod" }{TEXT -1 80 " determines which proced ures are used for solving this system and can be set to " }{TEXT 19 4 "auto" }{TEXT -1 2 ", " }{TEXT 19 3 "ABP" }{TEXT -1 2 ", " }{TEXT 19 9 "gausselim" }{TEXT -1 4 " or " }{TEXT 19 5 "solve" }{TEXT -1 13 ", d efault is " }{TEXT 19 4 "auto" }{TEXT -1 78 ".\nIf solve is specified, the builtin Maple procedure solve is used. If set to " }{TEXT 19 9 "g ausselim" }{TEXT -1 91 ", Gaussian elimination is used (first proposed by P. Paule and A. Riese, see References in " }{HYPERLNK 17 "qsum" 2 "qsum" "" }{TEXT -1 93 "). Using ABP, an algorithm of Abramov, Bronste in and Petkovsek is applied (see References in " }{HYPERLNK 17 "qsum" 2 "qsum" "" }{TEXT -1 33 "). The value auto applies either " }{TEXT 19 3 "ABP" }{TEXT -1 4 " or " }{TEXT 19 9 "gausselim" }{TEXT -1 78 ", \+ depending on the example. In most cases it should choose the fastest m ethod." }}{PARA 1T'{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 14 5 } {PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 'ame, the summation vari able\nx - a name, the " }{TEXT 278 1 "q" }{TEXT 279 20 "-difference equation" }{TEXT -1 29 " variable\ns - a name, the " }{TEXT 280 1 "q" }{TEXT 281 20 "-difference equation" }{TEXT -1 193 " function\na, \+ b - algebraic expressions, upper and lower summation bounds\nupper, lo wer - lists of algebraic expressions, the upper and the lower\n \+ parameters of the general " }{TEXT 268 1 "q" }{TEXT -1 160 "-hypergeometric function\nqq - a name or a name^integer\nz \+ - an algebraic expression, the evaluation point\n... - a sequence o f equations, optional arguments" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 42 "This function is an \+ implementation of the " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 51 "-an alogue of Zeilberger's algorithm, calculating a " }{TEXT 283 1 "q" } {TEXT -1 32 "-difference equation for the sum" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "s(x) = sum(f(x,k),k = -infinity .. infinity);" "6#/-%\ E "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Function:" }{TEXT 257 25 " qhyperterm - Produces a " }{XPPEDIT 18 0 "q" "6#%\"qG" }{Tb 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 14 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT 257 29 " Customizing the qsum package" }}{PARA 4 "" 0 "" {TEXT 26 13 "Introduc tion:" }}{PARA 0 "" 0 "" {TEXT -1 23 "Some procedures in the " }{TEXT 259 4 "qsum" }{TEXT -1 108 " package have optional arguments to contro l their behavior. Those optional arguments are always of the type " } {TEXT 19 9 "name=name" }{TEXT -1 290 ", where the left-hand side is th e name of the option and the right-hand side is the corresponding valu e. This is a good way to change the default setting for a single compu tation. If you want to use a different setting for many computations y ou can also change the default setting globally." }}{PARA 0 "" 0 "" {TEXT -1 255 "Associated with every optional argument in the qsum pack age is a corresponding global variable. Those global variables always \+ start with `_`. Usually it [F(F0F(-F+6#,&F'F(F(F(F(F(*&,(F/F(F(F(F'F(F(-F+6#F'F(F(\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "GFdiffeq((1-z)^(-alp ha-1)*exp(x*z/(z-1)),1,z,n,Laguerre(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"xG\"\"\"-%%diffG6$-%)LaguerreG6#F&-%\"$G6$F&\"\"#F'F'*&, (F&F'%&alphaG!\"\"F'F5F'-F)6$F+F&F'F5*&F+F'%\"nGF'F'\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum " 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "contgosper" 2 "contgospe r" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intrecursion" 2 "intrecursion" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "rodriguesrecursion and rodriguesdiffeq" 2 "ro drigues" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffrule and sumintrule " 2 "sumdiffrule" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumrecursion and \+ zeilberger" 2 "sumrecursion" "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } \"#F(F(-%)LaguerreG6#F&F(F(*&,*%&alphaG!\"\"%\"xGF(\" \"$F0*&F)F(F'tvolrotacambridglondonyorkoekoekswarttouwaskeschemorthogonalfacultinformattechnologsystemdelftkoornwindrigorouapplpaulriesmathematicaalgebraicalmotivatedtelescopfieldinstitutcommunicatproceedworkshopfunctionrelattopicjunetorontotarioismailetalamericalmathematicalsocietrovidencriescriptqsumimplementatqganalogugospzeilbergshortdescriptiontheorpetkovsekalgorithmhebasedhypergeometriccasedescribsummatalgorithmicapproachspecialfunctidentitiviewegbraunschweigwiesbadenisbnproceduresintroducbookprovidmaplhsummplsomeprocdurealsoavailablsumtoolfunctionqbinomialqbracketqfactorialqgammaqpochhammqfunctqhypertermqpsihypertermqratiosimpcombqsimpcombqsimplifqgosposperqsumrecursqsumrecurionqsumdiffeqqrecsolvsumqhypmaycustomglobalvariablsettreferencabramovbronsteinpetkovsekpolynomialsolutionlinearoperatorequationleveltedissacacmpresnewyorkgasprahmanbasicseriencyclopediamathematicapplica {VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal " -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 -" }{XPPEDIT 18 0 "q" "6#%\"qG" } {TEXT -1 82 "-analogue: a rigorous description. J. of Comput. and Appl . Math. 48, 1993, 91-111." }}{PARA 15 "" 0 "" {TEXT -1 39 "Paule, P. u nd Riese, A.: A Mathematica " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 84 "-analogue of Zeilberger's algorithm based on an algebraically moti vated approach to " }{TEXT 262 1 "q" }{TEXT -1 122 "-hypergeometric te lescoping. In: Fields Institute Communications, Vol. 14, Proceedings o f the Workshop Special Functions, " }{XPPEDIT 18 0 "q" "6#%\"qG" } {TEXT -1 155 "-Series and Related Topics, 12-23 June 1995, Toronto, On tario. Ed. by M. E. H. Ismail et al., Americal Mathematical Society, P rovidence, RI, 1997, 179-210." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } IT 18 0 "q;" "6#%\"qG" }{TEXT -1 109 "-analogue. Report 98 -17, Faculty of Information Technology and Systems, Delft University o f Technology, 1998." }}{PARA 15 "" 0 "" {TEXT -1 54 "Koornwinder, T. H .: On Zeilberger's algorithm and its "F(F0F(-F+6#,&F'F(F(F(F(F(*&,(F/F(F(F(F'F(F(-F+6#F'F(F(\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "GFdiffeq((1-z)^(-alp ha-1)*exp(x*z/(z-1)),1,z,n,Laguerre(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"xG\"\"\"-%%diffG6$-%)LaguerreG6#F&-%\"$G6$F&\"\"#F'F'*&, (F&F'%&alphaG!\"\"F'F5F'-F)6$F+F&F'F5*&F+F'%\"nGF'F'\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "seealso" {TEXT 26 10 "See Also: " }{HYPERLNK 17 "hsum " 2 "hsum" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "contgosper" 2 "contgospe r" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "intrecursion" 2 "intrecursion" " " }{TEXT -1 2 ", " }{HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "rodriguesrecursion and rodriguesdiffeq" 2 "ro drigues" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumdiffrule and sumintrule " 2 "sumdiffrule" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sumrecursion and \+ zeilberger" 2 "sumrecursion" "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } \"#F(F(-%)LaguerreG6#F&F(F(*&,*%&alphaG!\"\"%\"xGF(\" \"$F0*&F)F(F'#" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "rea d `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeo metric~Summation\",~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfram~Koepf,~University~of~KasselG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Generalized Laguerre Polynomials \+ " }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#%\"xG" }{TEXT -1 26 " wi th generating function:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "sum(f[n](x )*z^n,n=0..infinity)=(1-z)^(-alpha-1)*exp(x*z/(z-1))" "6#/-%$sumG6$*&- &%\"fG6#%\"nG6#%\"xG\"\"\")%\"zGF,F//F,;\"\"!%)infinityG*&),&F/F/F1!\" \",&%&alphaGF9F/F9F/-%$expG6#*(F.F/F1F/,&F1F/F/F9F9F/" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 37 "recurrence and differential equation: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "GFrecursion((1-z)^(-alpha-1)*ex p(x*z/(z-1)),1,z,Laguerre(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(* &,&%\"nG\"\"\"\"\"#F(F(-%)LaguerreG6#F&F(F(*&,*%&alphaG!\"\"%\"xGF(\" \"$F0*&F)F(F'#EDIT 18 0 "F(z)=sum(a[n]*f[n](x)*z^n,n=0..infini ty)" "6#/-%\"FG6#%\"zG-%$sumG6$*(&%\"aG6#%\"nG\"\"\"-&%\"fG6#F/6#%\"xG F0)F'F/F0/F/;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 70 "The procedure GFrecursion/GFdiffeq is an application of t he procedure " }{HYPERLNK 17 "intrecursion" 2 "intrecursion" "" } {TEXT -1 1 "/" }{HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 6 " thus " }{XPPEDIT 18 0 "F(z)" "6#-%\"FG6#%\"zG" }{TEXT -1 33 " has to be hyperexponential wrt. " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 1 " ." }}{PARA 15 "" 0 "" {TEXT -1 90 "The output of GFrecursion/GFdiffeq \+ is a recurrence/differential equation for the function " }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#%\"xG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 79 "The procedures will look for a recurrence/differential equation of order up to " }{TEXT 256 8 "MAXORDER" }{TEXT -1 107 ". Th is global variable is set by default to 5 but may be changed to any po sitive integer value or infinity." }}}{SECT 0 {PARA 0 "$ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormallistitembulletfunctintrecursalgorithmalmkvistzeilbergdeterminholonomicrecurrencequatdefinitintegralusagcallsequencintrecurionparameterexpressnameintegratvariablsynopsidescriptimplementattrierecurrenceintsgngintgfgtgagbgassuminghyperexponentialwrthypergometriccontinuouverssumrecurssumrecursionremembtermggratiodiffdiffgtgfrationalenhanccontgospapplifunctisumsigmasumgsigmagjgthuscalculatantiderivatsuchiffgoverwegetresultbgfwarnalgorithmrighthandsideabovequatiovanishreturnhomogeneouprocedurwilllookordeupmaxordglobalsetdefaultbutmaychanganypositintegvalueinfinitpossiblobtainrationalcertificathsumtheorallowsimplposterioproofusingoptionalargumtruewilwithentrfirstcomputsecondcertificatexamplreadhsumplvpackaghypergeometricsummatcopyrightwolframkoepfuniversitkasselggammagammagfunctiontsimpfuncsimpl;simpledsimpler;(  simplifsimplifi  simplificat? simplifictsimplifysin#sinc singibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalbulletitemlistfunctgfrecursgfdiffeqcalculatrecurrencdifferentialequationgivengeneratusagecallsequencparameterexpressnamesynopsidescriptthesfunctiondetermindifferentialfamilifgngxgzgwithsuminfinitysumgaginfinitygprocedurapplicatheintrecursintdiffeqthushyperexponentialwrtwilllookorderupmaxordthglobalvariablsetdefaultbutmaychanganypositivintegvaluinfinitexamplreahsummplvpackaghypergeometricsummatycopyrightwolframkoepfuniversitkasselggenerallaguerrpolynomialwialphaexpzgfalphagfexpgexlaguerregalphagxgfalphadiffgngfseealsoalsocontgosprodriguesrecursrodriguesdiffeqrodrigusumdiffrulsumintrulsumrecurszeilberg{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "Bullet Item" -1 15 1 :" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "rea d `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeo metric~Summation\",~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2002,~Wolfram~Koepf,~University~of~KasselG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Generalized Laguerre Polynomials \+ " }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#%\"xG" }{TEXT -1 26 " wi th generating function:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "sum(f[n](x )*z^n,n=0..infinity)=(1-z)^(-alpha-1)*exp(x*z/(z-1))" "6#/-%$sumG6$*&- &%\"fG6#%\"nG6#%\"xG\"\"\")%\"zGF,F//F,;\"\"!%)infinityG*&),&F/F/F1!\" \",&%&alphaGF9F/F9F/-%$expG6#*(F.F/F1F/,&F1F/F/F9F9F/" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 37 "recurrence and differential equation: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "GFrecursion((1-z)^(-alpha-1)*ex p(x*z/(z-1)),1,z,Laguerre(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(* &,&%\"nG\"\"\"\"\"#F(F(-%)LaguerreG6#F&F(F(*&,*%&alphaG!\"\"%\"xGF(\" \"$F0*&F)F(F'EDIT 18 0 "F(z)=sum(a[n]*f[n](x)*z^n,n=0..infini ty)" "6#/-%\"FG6#%\"zG-%$sumG6$*(&%\"aG6#%\"nG\"\"\"-&%\"fG6#F/6#%\"xG F0)F'F/F0/F/;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 70 "The procedure GFrecursion/GFdiffeq is an application of t he procedure " }{HYPERLNK 17 "intrecursion" 2 "intrecursion" "" } {TEXT -1 1 "/" }{HYPERLNK 17 "intdiffeq" 2 "intdiffeq" "" }{TEXT -1 6 " thus " }{XPPEDIT 18 0 "F(z)" "6#-%\"FG6#%\"zG" }{TEXT -1 33 " has to be hyperexponential wrt. " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 1 " ." }}{PARA 15 "" 0 "" {TEXT -1 90 "The output of GFrecursion/GFdiffeq \+ is a recurrence/differential equation for the function " }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#%\"xG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 79 "The procedures will look for a recurrence/differential equation of order up to " }{TEXT 256 8 "MAXORDER" }{TEXT -1 107 ". Th is global variable is set by default to 5 but may be changed to any po sitive integer value or infinity." }}}{SECT 0 {PARA 0 "+} 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "List Item" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 3 3 1 0 1 0 2 2 14 5 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 114 "GFrecursion, GFdiffeq - Calculate a recurrence/differential equat ion for a function given by a generating function" }}{PARA 0 "" 0 "usa ge" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 55 "\n GFrecursion(F,a ,z,s(n));\n GFdiffeq(F,a,z,n,s(x));" }}{PARA 0 "" 0 "" {TEXT 26 11 " Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 6 "F,a - " }{TEXT -1 17 "an expression\n " }{TEXT 23 10 "n,s,x,z - " }{TEXT -1 6 "a name" }}} {SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 88 "These functions determine a recurrence/differenti al equation for a familiy of functions " }{XPPEDIT 18 0 "f[n](x)" "6#- &%\"fG6#%\"nG6#%\"xG" }{TEXT -1 35 ", given by the generating function " }{XPPEDIT 18 0 "F(z)" "6#-%\"FG6#%\"zG" }{TEXT -1 5 " with" }} {PARA 257 "" 0 "" {XPP,7ibmintelntmathtimehyperlinkcommhelpheadnormaltimeslistitembulletintroductqsumpackagcopyrightharaldingwolframkoepfuniversitykasselcommentbugreportrewelcomsentsecondauthorviamailprofdruniversitndepartmematiccomputsciencnheinrichplettstrndmathematikunideurlhttpwwwdescriptqsumimplementatqganalogugospzeilbergshortdescriptiontheorpetkovsekalgorithmhebasedhypergeometriccasedescribsummatalgorithmicapproachspecialfunctidentitiviewegbraunschweigwiesbadenisbnproceduresintroducbookprovidmaplhsummplsomeprocdurealsoavailablsumtoolfunctionqbinomialqbracketqfactorialqgammaqpochhammqfunctqhypertermqpsihypertermqratiosimpcombqsimpcombqsimplifqgosposperqsumrecursqsumrecurionqsumdiffeqqrecsolvsumqhypmaycustomglobalvariablsettreferencabramovbronsteinpetkovsekpolynomialsolutionlinearoperatorequationleveltedissacacmpresnewyorkgasprahmanbasicseriencyclopediamathematicapplica Zibmintelntmathtimecommhelpheadnormallistitembulletqganalogugospalgorithmanaloguedecisprocedurdecidwhethhypergeometrictermfgkgantidifferencggreducproblemfindsuchantidifferencefinitlaurpolynomialsatisffirstorderrecurrencequatmainstepcalculatrepresentatrationalfunctpgrgwithiigcdgcdgjgfallnonnegatintegerjgxgsolvinhomogeneoudeterminlowerupperdegreboundsubstitutgenerichesystemlinearequationesultcomparcoefficientmonomialnosolutwecanconcludexisthypergometricnoteratiozeilbergdealdefnitsummathypergeomtricngrespecttrierecurrencsumsggivensumgagbgideadeducsummandapplysigmasigmagyetundeterminigmapositintegereasyshowalogudeliversystemabovthusinsteadonlyalsogetequationkgfmultiplreallysimplprovposterioriindependderivatjustdividverifresultcallcertificatequatisummoveralphaalphagmaxbetabetagminobtainlphaetafofdflsupportaturalinfinitinfinitygpartanishq hypergeomg  hyperlink@hypert  hyperterm7* hypertermgial7 ibm5ic?icatideidea identitidentitiiedifferenciffg igigf igmaigmag  iguesrecurs iiiiiilateralileimalimesimpcombimplem implement implementa implementat_ implementatioimpler includincreasind}}{PARA 0 "" 0 "" {TEXT -1 14 " ratio(f,k);" }}{PARA 0 "" 0 " " {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 4 "f - " } {TEXT -1 13 "an expression" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 387 "The function sim pcomb(f) simplifies an expression f involving powers, factorials, GAMM A function terms, binomial coefficients, and Pochhammer symbols by con verting factorials, binomial coefficients, and Pochhammer symbols into GAMMA function terms, and simplifying its result. It is designed to d etect rationality. If the output is not rational, it is given in terms of GAMMA functions. " }}{PARA 15 "" 0 "" {TEXT -1 144 "The function r atio(f,k) simplifies the ratio f(k+1)/f(k) using simpcomb. The functio n simpcomb is designed to detect rationality of such ratios." }}} {SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~SuXxtend ycopyrightoyet ygfyieldyor york youGyperexponential ypergeometric zcertificat zeilzeilbzeilbergoV    zerozfzg?#zgf' (see " }{HYPERLNK 17 "qsum[setting]" 2 "qsum[setting]" "" } {TEXT -1 2 ")." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "References:" } }{PARA 15 "" 0 "" {TEXT -1 123 "Abramov, S. A., Bronstein, M. and Petk ovsek, M.: On polynomial solutions of linear operator equations, In T. Levelt (ed.), " }{TEXT 257 15 "Proc. ISSAC '95" }{TEXT -1 37 ", 190-2 96. ACM Press, New York, 1995." }}{PARA 15 "" 0 "" {TEXT -1 27 "Gasper , G. and Rahman, M.: " }{TEXT 263 27 "Basic Hypergeometric Series" } {TEXT -1 134 ". Encyclopedia of Mathematics and its Applications, Vol. 35. Ed. by G.-C. Rota, Cambridge University Press, London and New Yor k, 1990." }}{PARA 15 "" 0 "" {TEXT -1 100 "Koekoek, R. and Swarttouw, \+ F.F.: The Askey-scheme of hypergeometric orthogonal polynomials and it s -" }{XPPEDIT 18 0 "q;" "6#%\"qG" }{TEXT -1 109 "-analogue. Report 98 -17, Faculty of Information Technology and Systems, Delft University o f Technology, 1998." }}{PARA 15 "" 0 "" {TEXT -1 54 "Koornwinder, T. H .: On Zeilberger's algorithm and its "{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "List Item" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 6 6 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 37 " \+ Introduction to the qsum package " }}{PARA 256 "" 0 "" {TEXT 256 75 "Copyright 1998-2002 by Harald B\366ing and Wolfram Koepf, Universi ty of Kassel" }}{PARA 0 "" 0 "" {TEXT -1 96 "Comments or bug reports a re welcome and should be sent to the second author via mail or e-mail: \n" }}{PARA 258 "" 0 "" {TEXT -1 24 "Prof. Dr. Wolfram Koepf\n" } {TEXT 259 20 "University of Kassel" }{TEXT -1 94 "\nDepartment of Math ematics and Computer Science\nHeinrich-Plett-Str. 40\nD-34132 Kassel\n e-mail: " }{TEC Maple package " } {HYPERLNK 17 "hsum.mpl" 2 "hsum" "" }{TEXT -1 52 ". (Some of the proce dures are also available in the " }{HYPERLNK 17 "sumtools" 2 "sumtools " "" }{TEXT -1 10 " package)." }}{PARA 15 "" 0 "" {TEXT -1 49 "The fun ctions available in qsum6.mpl are:\n " }{HYPERLNK 17 "qbinomial, qbrackets, qfactorial, qGAMMA and qpochhammer" 2 "qfunctions" "" } {TEXT -1 9 ",\n " }{HYPERLNK 17 "qhyperterm and qpsihyperterm" 2 "qhyperterm" "" }{TEXT -1 9 ",\n " }{HYPERLNK 17 "qratio and q simpcomb" 2 "qsimpcomb" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qsimplify" 2 "qsimplify" "" }{TEXT -1 8 "\n " }{HYPERLNK 17 "qgosper" 2 "qg osper" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qsumrecursion" 2 "qsumrecurs ion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qsumdiffeq" 2 "qsumdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qrecsolve" 2 "qrecsolve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sum2qhyper" 2 "sum2qhyper" "" }{TEXT -1 1 "." }} {PARA 15 "" 0 "" {TEXT -1 56 "The package may be customized via global variables 6XT 258 31 "koepf@mathematik.uni-kassel.de\n" }{TEXT 260 4 "URL:" }{TEXT 261 43 " http://www.mathematik.uni-kassel.de/~koepf" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 13 "Description: " }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 45 "The qs um package is an implementation of the " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 66 "-analogues of Gosper's, Zeilberger's (for a short descri ption see " }{HYPERLNK 17 "qsum[theory]" 2 "qsum[theory]" "" }{TEXT -1 28 ") and Petkovsek's algorithm." }}{PARA 15 "" 0 "" {TEXT -1 67 "T he implementation is based on the hypergeometric case described in" }} {PARA 257 "" 0 "" {TEXT -1 13 "Wolfram Koepf" }}{PARA 257 "" 0 "" {TEXT -1 24 "Hypergeometric Summation" }}{PARA 257 "" 0 "" {TEXT -1 69 "An Algorithmic Approach to Summation and Special Function Identiti es." }}{PARA 257 "" 0 "" {TEXT -1 57 "Vieweg, Braunschweig/Wiesbaden, \+ 1998, ISBN 3-528-06950-3." }}{PARA 14 "" 0 "" {TEXT -1 72 "The procedu res introduced in the book are provided by the8{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "List Item" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 6 6 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help For:" }{TEXT -1 37 " \+ Introduction to the qsum package " }}{PARA 256 "" 0 "" {TEXT 256 75 "Copyright 1998-2002 by Harald B\366ing and Wolfram Koepf, Universi ty of Kassel" }}{PARA 0 "" 0 "" {TEXT -1 96 "Comments or bug reports a re welcome and should be sent to the second author via mail or e-mail: \n" }}{PARA 258 "" 0 "" {TEXT -1 24 "Prof. Dr. Wolfram Koepf\n" } {TEXT 259 20 "University of Kassel" }{TEXT -1 94 "\nDepartment of Math ematics and Computer Science\nHeinrich-Plett-Str. 40\nD-34132 Kassel\n e-mail: " }{TE9ibmintelntmathtimehyperlinkcommhelpheadnormalbulletitemtimesdashlistintroducthsumpackagusagcopyrightwolframkoepfuniversitkasselyoucancontactauthorviamailprofdrndepartmematiccomputsciencnheinrichplettstrndmathematikunideurlhttpwwwsynopsidescriptcollectroutinwrittenwereintroducbookhypergeometricsummatalgorithmicapproachspecialfunctidentitviewegbraunschweigwiesbadenshortreferencfoundatcompletbibliographabovsomepackagalreadimplementmaplreleasfurthinformatsumtoolsumtoolqganaloguproceduravailabqsumintroductgospzeilbergalgorithmtheorfollowfunctionavailablsimpcombimpcombhypertermsumtohypsumtohypkfreerecfasenmyextendextendsumrecursclosedformzcertificatwzcertificatsumdeltanablasumdeltanablarecpolyrecpohyperrechypsumdiffeqsumdiffrulsumintrulsumdiffrulecontgosprecursintrecursintdiffeqintdiffeqrodriguesrecursrodriguesdiffeqgfrecursgfdiffseealalso ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpeadingnormalheadlistitembulletcustomizqsumpackagintroductionsomeprocedurhaveoptionalargumentcontrotheirbehaviorthosalwaytypenamelefthandsidethoptionrightcorrespondvalugoodwaychangdefaultsettsinglcomputationyouwantusediffermanycomputatoucanalsoglobalassociatwitheverargumpackagevariablstartusualfollowanothfinalqsumrecursrecrangthusrecrangusedmoremostcaseassignosesamepossibloptiodescriptvariableqgospqsumrecursiconsumpartconsistsolvsystemlinearequationheortheorsolvemethoddeterminprocuressetautoabpgausselimefaultnifspecifibuiltinausselimgaussianeliminatfirstpropospaulriesreferencusingalgorithmabramovbronstpetkovsekappliappleithdependexamplchoosfastestethodfactornormalexpandotherreturngausseliminatnwhenequationssimplificatproblemdefaultseemhowevsometiminsteadyieldbettresultspecialsolutmaytruefs{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT 26 32 "-analogue of Gosper's algorithm:" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 79 "-analo gue of Gosper's algorithm is a decision procedure that decides whether a " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 21 "-hypergeometric term \+ " }{XPPEDIT 18 0 "F(k)" "6#-%\"FG6#%\"kG" }{TEXT -1 9 " (w.r.t. " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 8 ") has a " }{XPPEDIT F" }{XPPEDIT 18 0 "q" "6#%\"qG" } {TEXT -1 82 "-analogue: a rigorous description. J. of Comput. and Appl . Math. 48, 1993, 91-111." }}{PARA 15 "" 0 "" {TEXT -1 39 "Paule, P. u nd Riese, A.: A Mathematica " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 84 "-analogue of Zeilberger's algorithm based on an algebraically moti vated approach to " }{TEXT 262 1 "q" }{TEXT -1 122 "-hypergeometric te lescoping. In: Fields Institute Communications, Vol. 14, Proceedings o f the Workshop Special Functions, " }{XPPEDIT 18 0 "q" "6#%\"qG" } {TEXT -1 155 "-Series and Related Topics, 12-23 June 1995, Toronto, On tario. Ed. by M. E. H. Ismail et al., Americal Mathematical Society, P rovidence, RI, 1997, 179-210." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } IT 18 0 "q;" "6#%\"qG" }{TEXT -1 109 "-analogue. Report 98 -17, Faculty of Information Technology and Systems, Delft University o f Technology, 1998." }}{PARA 15 "" 0 "" {TEXT -1 54 "Koornwinder, T. H .: On Zeilberger's algorithm and its ?(see " }{HYPERLNK 17 "qsum[setting]" 2 "qsum[setting]" "" } {TEXT -1 2 ")." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "References:" } }{PARA 15 "" 0 "" {TEXT -1 123 "Abramov, S. A., Bronstein, M. and Petk ovsek, M.: On polynomial solutions of linear operator equations, In T. Levelt (ed.), " }{TEXT 257 15 "Proc. ISSAC '95" }{TEXT -1 37 ", 190-2 96. ACM Press, New York, 1995." }}{PARA 15 "" 0 "" {TEXT -1 27 "Gasper , G. and Rahman, M.: " }{TEXT 263 27 "Basic Hypergeometric Series" } {TEXT -1 134 ". Encyclopedia of Mathematics and its Applications, Vol. 35. Ed. by G.-C. Rota, Cambridge University Press, London and New Yor k, 1990." }}{PARA 15 "" 0 "" {TEXT -1 100 "Koekoek, R. and Swarttouw, \+ F.F.: The Askey-scheme of hypergeometric orthogonal polynomials and it s -" }{XPPEDIT 18 0 "q;" "6#%\"qG" }{TEXT -1 109 "-analogue. Report 98 -17, Faculty of Information Technology and Systems, Delft University o f Technology, 1998." }}{PARA 15 "" 0 "" {TEXT -1 54 "Koornwinder, T. H .: On Zeilberger's algorithm and its ? Maple package " } {HYPERLNK 17 "hsum.mpl" 2 "hsum" "" }{TEXT -1 52 ". (Some of the proce dures are also available in the " }{HYPERLNK 17 "sumtools" 2 "sumtools " "" }{TEXT -1 10 " package)." }}{PARA 15 "" 0 "" {TEXT -1 49 "The fun ctions available in qsum6.mpl are:\n " }{HYPERLNK 17 "qbinomial, qbrackets, qfactorial, qGAMMA and qpochhammer" 2 "qfunctions" "" } {TEXT -1 9 ",\n " }{HYPERLNK 17 "qhyperterm and qpsihyperterm" 2 "qhyperterm" "" }{TEXT -1 9 ",\n " }{HYPERLNK 17 "qratio and q simpcomb" 2 "qsimpcomb" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qsimplify" 2 "qsimplify" "" }{TEXT -1 8 "\n " }{HYPERLNK 17 "qgosper" 2 "qg osper" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qsumrecursion" 2 "qsumrecurs ion" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qsumdiffeq" 2 "qsumdiffeq" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qrecsolve" 2 "qrecsolve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "sum2qhyper" 2 "sum2qhyper" "" }{TEXT -1 1 "." }} {PARA 15 "" 0 "" {TEXT -1 56 "The package may be customized via global variables @XT 258 31 "koepf@mathematik.uni-kassel.de\n" }{TEXT 260 4 "URL:" }{TEXT 261 43 " http://www.mathematik.uni-kassel.de/~koepf" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 13 "Description: " }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 45 "The qs um package is an implementation of the " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 66 "-analogues of Gosper's, Zeilberger's (for a short descri ption see " }{HYPERLNK 17 "qsum[theory]" 2 "qsum[theory]" "" }{TEXT -1 28 ") and Petkovsek's algorithm." }}{PARA 15 "" 0 "" {TEXT -1 67 "T he implementation is based on the hypergeometric case described in" }} {PARA 257 "" 0 "" {TEXT -1 13 "Wolfram Koepf" }}{PARA 257 "" 0 "" {TEXT -1 24 "Hypergeometric Summation" }}{PARA 257 "" 0 "" {TEXT -1 69 "An Algorithmic Approach to Summation and Special Function Identiti es." }}{PARA 257 "" 0 "" {TEXT -1 57 "Vieweg, Braunschweig/Wiesbaden, \+ 1998, ISBN 3-528-06950-3." }}{PARA 14 "" 0 "" {TEXT -1 72 "The procedu res introduced in the book are provided by theAic polynomial in t he recurrence equation and solve the system of linear equations that r esults by comparing the coefficients of all monomials in " }{XPPEDIT 18 0 "q^k" "6#)%\"qG%\"kG" }{TEXT -1 2 ".)" }}{PARA 15 "" 0 "" {TEXT -1 45 "If there is no solution we can conclude that " }{XPPEDIT 18 0 " F(k)" "6#-%\"FG6#%\"kG" }{TEXT -1 8 " has no " }{XPPEDIT 18 0 "q" "6#% \"qG" }{TEXT -1 31 "-hypergeometric antidifference." }}{PARA 15 "" 0 " " {TEXT -1 14 "If a solution " }{XPPEDIT 18 0 "X(k)" "6#-%\"XG6#%\"kG " }{TEXT -1 17 " exists the term " }{XPPEDIT 18 0 "G(k)" "6#-%\"GG6#% \"kG" }{TEXT -1 5 " with" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "G(k):= R (k-1)*X(k-1)/P(k)*F(k)" "6#>-%\"GG6#%\"kG**-%\"RG6#,&F'\"\"\"F-!\"\"F- -%\"XG6#,&F'F-F-F.F--%\"PG6#F'F.-%\"FG6#F'F-" }}{PARA 15 "" 0 "" {TEXT -1 5 "is a " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 35 "-hyperge ometric antidifference for " }{XPPEDIT 18 0 "F(k)" "6#-%\"FG6#%\"kG" } {TEXT -1 23 ". (Note that the ratio " }{XPPEDIT 18 0 "G(k)" "6#-%\"GG6 #%\"kG" }{TEXT -PEDIT 18 0 "F(k+1)/F(k) = P(k+1)/P(k) * Q(k)/ R(k)" "6#/*&-%\"FG6#,&%\"kG\"\"\"F*F*F*-F&6#F)!\"\"**-%\"PG6#,&F)F*F*F *F*-F06#F)F--%\"QG6#F)F*-%\"RG6#F)F-" }{TEXT -1 11 ",\n(ii) " } {XPPEDIT 18 0 "gcd(Q(k), R(k+j)) = 1" "6#/-%$gcdG6$-%\"QG6#%\"kG-%\"RG 6#,&F*\"\"\"%\"jGF/F/" }{TEXT -1 33 " , for all nonnegative integers \+ " }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 33 "Find a finite Laurent polynomial " }{XPPEDIT 18 0 "X(k)" "6#-%\"XG6#%\"kG" }{TEXT -1 5 " (in " }{XPPEDIT 18 0 "q^k" "6#)%\"qG% \"kG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q^(-k)" "6#)%\"qG,$%\"kG!\" \"" }{TEXT -1 94 ") that solves the inhomogeneous recurrence equation \+ of first order\n(1) " }{XPPEDIT 18 0 "P(k) = \+ Q(k)*X(k) - R(k-1)*X(k-1)" "6#/-%\"PG6#%\"kG,&*&-%\"QG6#F'\"\"\"-%\"XG 6#F'F-F-*&-%\"RG6#,&F'F-F-!\"\"F--F/6#,&F'F-F-F6F-F6" }{TEXT -1 46 "\n (Determine lower and upper degree bounds for " }{XPPEDIT 18 0 "X(k)" " 6#-%\"XG6#%\"kG" }{TEXT -1 165 ", substitute a generD18 0 "q" "6#%\"qG" }{TEXT -1 31 "-hypergeometric antidifference " }{XPPEDIT 18 0 "G(k)" "6#-%\"GG6#%\"kG" }{TEXT -1 6 ", i.e." }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "F(k) = G(k+1) - G(k)" "6#/-%\"FG6#%\"kG,&-%\"GG6#,&F'\" \"\"F-F-F--F*6#F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 176 "The algorithm reduces the problem of finding such an antidifferen ce to finding a finite Laurent polynomial that satisfies a first order recurrence equation. The main steps are:" }}{PARA 15 "" 0 "" {TEXT -1 12 "Calculate a " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 49 "-Gospe r representation for the rational function " }{XPPEDIT 18 0 "F(k+1)" " 6#-%\"FG6#,&%\"kG\"\"\"F(F(" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "F(k)" "6# -%\"FG6#%\"kG" }{TEXT -1 19 ", i.e. polynomials " }{XPPEDIT 18 0 "P(k) " "6#-%\"PG6#%\"kG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Q(k)" "6#-%\"QG6# %\"kG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "R(k)" "6#-%\"RG6#%\"kG" } {TEXT -1 4 " in " }{XPPEDIT 18 0 "q^k" "6#)%\"qG%\"kG" }{TEXT -1 17 " \+ with:\n(i) " }{XPEsolv/#solvem solvemethodsome'sometimsort soughtsparc;specispecial#  specialsolutspecifspecifi'specify7*{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bul let Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } 0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }>contain explicit summation bou nds which are not equal to " }{TEXT 19 19 "-infinity..infinity" } {TEXT -1 6 ". (If " }{TEXT 19 6 "always" }{TEXT -1 4 " or " }{TEXT 19 5 "never" }{TEXT -1 66 " is specified, the right-hand side is always o r never calculated.)" }}{PARA 15 "" 0 "" {TEXT -1 20 "The global varia ble " }{TEXT 19 26 "_qsumrecursion_inhomo2homo" }{TEXT -1 15 " can be \+ set to " }{TEXT 19 4 "true" }{TEXT -1 4 " or " }{TEXT 19 5 "false" } {TEXT -1 13 ", default is " }{TEXT 19 5 "false" }{TEXT -1 16 ".\nIf th e option " }{TEXT 19 3 "rhs" }{TEXT -1 11 " is set to " }{TEXT 19 4 "t rue" }{TEXT -1 63 ", the resulting recurrence equation might be inhomo geneous. If " }{TEXT 19 11 "inhomo2homo" }{TEXT -1 11 " is set to " } {TEXT 19 4 "true" }{TEXT -1 93 ", qsumrecursion will convert this recu rrence equation into a homogeneous one of higher order." }}{PARA 15 " " 0 "" {TEXT -1 20 "The global variable " }{TEXT 19 26 "_qsumrecursion _certificate" }{TEXT -1 15 " can be set to " }{TEXT 19 4 "down" } {TEXT -nteger value (symbolic). The function is defined as: " }} {PARA 257 "" 0 "" {XPPEDIT 18 0 "qpochhammer(a,q,infinity)=product(1-a *q^j,j=0..infinity)" "6#/-%,qpochhammerG6%%\"aG%\"qG%)infinityG-%(prod uctG6$,&\"\"\"F.*&F'F.)F(%\"jGF.!\"\"/F1;\"\"!F)" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 2 "6#/-%,qpochhamm erG6%%\"aG%\"qG%\"kG-%*PIECEWISEG6%7$-%(productG6$,&\"\"\"F2*&F'F2)F(% \"jGF2!\"\"/F5;\"\"!,&F)F2F6F22F9F)7$F2/F)F97$-F/6$*$F1F6/F5;F)F22F)F9 " }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 5 "qfac:" }{TEXT -1 81 " This f unction is a synonym for qpochhammer, i.e. qfac(a,q,k)=qpochhammer(a,q ,k)." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "qbrackets:" }{TEXT -1 105 " The qbrackets function has two arguments. Both can be any valid \+ algebraic expression. The definition is:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "qbrackets(k,q) = (q^k-1)/(q-1)" "6#/-%*qbracketsG6$%\"k G%\"qG*&,&)F(F'\"\"\"F,!\"\"F,,&F(F,F,F-F-" }}}{SECT 0 {PARA 0 "" 0 " " {TEXT 26 10 "qbinomial:" }{TEXT -1 233 " The qbi  57 " may be set to a posint or a range of posint, defa ult is " }{TEXT 19 4 "1..5" }{TEXT -1 120 ".\nIf a range is specified, qsumrecursion will start to look for a recurrence equation in that ra nge. A positive integer " }{TEXT 19 1 "n" }{TEXT -1 18 " is equivalent to " }{TEXT 19 4 "n..n" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 19 "The global variable" }{TEXT 19 18 "_qsumrecursion_rhs" }{TEXT -1 15 " can be set to " }{TEXT 19 6 "always" }{TEXT -1 2 ", " }{TEXT 19 8 "explicit" }{TEXT -1 4 " or " }{TEXT 19 5 "never" }{TEXT -1 303 ", d efault is explicit.\nIt determines whether for the conversion of the r ecurrence equation for the input term into one for the corresponding s um is done by calculating the right-hand side, i.e. the inhomogeneous \+ part of the recurrence equation, or just by assuming finite support of the input term (see " }{HYPERLNK 17 "qsum[theory]" 2 "qsum[theory]" " " }{TEXT -1 150 "). By default the right-hand side will be calculated \+ only if the parameters of qsumrecursion J useful for a \+ " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 72 "-hypergeometric expressio n containing an additional parameter such that " }{TEXT 19 7 "qgosper " }{TEXT -1 9 " finds a " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 90 "- hypergeometric antidifference whenever we substitute the parameter by \+ a positive integer." }}}{SECT 0 {PARA 4 "" 0 "" {HYPERLNK 17 "qsumrecu rsion" 2 "qsumrecursion" "" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 20 "The global variable " }{TEXT 19 24 "_qsumrecursion_recursion" } {TEXT -1 15 " may be set to " }{TEXT 19 4 "down" }{TEXT -1 4 " or " } {TEXT 19 2 "up" }{TEXT -1 13 ", default is " }{TEXT 19 4 "down" } {TEXT -1 245 ".\nIt controls whether the recurrence equation sought fo r by qsumrecursion is downwards or upwards. (We recommend to use the d efault value as switching to up leads to another linear system of equa tions. This seems to be slower for some examples.)" }}{PARA 15 "" 0 " " {TEXT -1 20 "The global variable " }{TEXT 19 23 "_qsumrecursion_reco rder" }{TEXT -1L" "6#-%\"fG6#%\"kG" }{TEXT -1 3 " if" }}{PARA 257 " " 0 "" {XPPEDIT 18 0 "f(k) = g(k+1) - g(k)" "6#/-%\"fG6#%\"kG,&-%\"gG6 #,&F'\"\"\"F-F-F--F*6#F'!\"\"" }{TEXT -1 20 " or " } {XPPEDIT 18 0 "f(k) = g(k) - g(k-1)" "6#/-%\"fG6#%\"kG,&-%\"gG6#F'\"\" \"-F*6#,&F'F,F,!\"\"F0" }}{PARA 14 "" 0 "" {TEXT -1 13 "respectively. " }}{PARA 15 "" 0 "" {TEXT -1 20 "The global variable " }{TEXT 19 17 " _qgosper_simplify" }{TEXT -1 15 " can be set to " }{TEXT 19 4 "NULL" } {TEXT -1 96 " or the name of any procedure, which will be used to simp lify the result of qgosper. Default is " }{TEXT 19 6 "factor" }{TEXT -1 54 ", i.e. the result of the procedure will be factorized." }} {PARA 15 "" 0 "" {TEXT -1 3 "If " }{TEXT 19 7 "qgosper" }{TEXT -1 60 " returns no antidifference, you might try to use the option " }{TEXT 19 11 "series=true" }{TEXT -1 16 ", thus allowing " }{TEXT 19 7 "qgosp er" }{TEXT -1 43 " to search for an antidifference that is a " }{TEXT 266 1 "q" }{TEXT -1 45 "-hypergeometric series. This isMversversionversit vertvivia+ viewegviiviiiviorvol vpackagsvsekwantwarnwarniway we3welcom wellwer were werserwgwgfwhwhenevwherwhereawheth# whilwi  wiesbadenwilwilf willW0wilson wilsont  wilsonterm wingwishwitwithUwithoutratiratioSrationrationarationalg? rationalitrcingrdrderre?8     rearead8bulletin butw"calcalculatc,callKcambridg can< 5 "" 0 "" {TEXT -1 20 "The global variable " }{TEXT 19 22 "_qsum_gausselim_normal" }{TEXT -1 15 " can be set to " }{TEXT 19 6 "factor" }{TEXT -1 2 ", " }{TEXT 19 6 "normal" }{TEXT -1 2 ", " } {TEXT 19 14 "normalexpanded" }{TEXT -1 43 " and other procedures that \+ are returned by " }{TEXT 19 30 "`gausseliminate/procedures`();" } {TEXT -1 13 ", default is " }{TEXT 19 6 "normal" }{TEXT -1 80 ".\nWhen Gaussian elimination is used for solving the system of linear equatio ns, " }{TEXT 19 22 "_qsum_gausselim_normal" }{TEXT -1 148 " determines which procedure is used for simplification. For most problems the def ault setting seems to be the fastest one. However, sometimes using " } {TEXT 19 6 "factor" }{TEXT -1 12 " instead of " }{TEXT 19 6 "normal" } {TEXT -1 23 " yields better results." }}{PARA 15 "" 0 "" {TEXT -1 20 " The global variable " }{TEXT 19 21 "_qsum_specialsolution" }{TEXT -1 15 " may be set to " }{TEXT 19 4 "true" }{TEXT -1 4 " or " }{TEXT 19 5 "false" }{TEXT -1 13 ", default is " }{TEXT 19 5T 26 9 "See also:" }{TEXT -1 1 " " }{HYPERLNK 17 " qfunctions" 2 "qfunctions" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qhyperte rm" 2 "qhyperterm" "" }{TEXT 257 2 ", " }{HYPERLNK 17 "qsimpcomb" 2 "q simpcomb" "" }{TEXT 258 2 ", " }{HYPERLNK 17 "qsum" 2 "qsum" "" } {TEXT 259 2 ", " }{HYPERLNK 17 "qsumrecursion" 2 "qsumrecursion" "" } {TEXT 260 2 ", " }{HYPERLNK 17 "qrecsolve" 2 "qrecsolve" "" }{TEXT 261 2 ", " }{HYPERLNK 17 "sum2qhyper" 2 "sum2qhyper" "" }{TEXT 262 2 " " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 } ormula (II.34)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "result:=qgosper((1-a*p^k*q^k)/(1-a )/c^k*qpochhammer(a,p,k)/\nqpochhammer(a*p/c,p,k)*qpochhammer(c,q,k)/ \nqpochhammer(q,q,k),[p,q],k=0..n);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%'resultG*4-%,qpochhammerG6%%\"cG%\"qG,&%\"nG\"\"\"F-F-F--F'6%%\"aG% \"pGF+F-,&F)!\"\"*&)F1F+F-F0F-F-F-,&F3F-)F*F+F-F--F'6%F*F*F+F3-F'6%*(F 0F-F1F-F)F3F1F+F3)F),&F,F3F3F-F-,&F3F-F0F-F3,&F3F-F)F-F3" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXU sum that \+ is a polynomial in " }{XPPEDIT 18 0 "q^k" "6#)%\"qG%\"kG" }{TEXT -1 11 " of degree " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "assume(n,posint);\nqgosper(term(n,k),q,k, series=true);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*0,&!\"\"\"\"\")%\"qG %\"kGF&F&%\"aGF&)F(%#n|irGF&-%$SumG6$**-%,qpochhammerG6%)F(,$F,F%F(%#_ iGF&,&*&F*F&F+F&F&)F(F6F%F%-F26%*&F*F%F4F&F(F6F%)F(*&F)F&F6F&F&/F6;\" \"!F,F&-F26%F*F(F)F&-F26%F(F(F)F%)F8,$F)F%F&" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 48 "Gasper, Rahman (1990), Appendix, formula (II.34)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "result:=qgosper((1-a*p^k*q^k)/(1-a )/c^k*qpochhammer(a,p,k)/\nqpochhammer(a*p/c,p,k)*qpochhammer(c,q,k)/ \nqpochhammer(q,q,k),[p,q],k=0..n);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%'resultG*4-%,qpochhammerG6%%\"cG%\"qG,&%\"nG\"\"\"F-F-F--F'6%%\"aG% \"pGF+F-,&F)!\"\"*&)F1F+F-F0F-F-F-,&F3F-)F*F+F-F--F'6%F*F*F+F3-F'6%*(F 0F-F1F-F)F3F1F+F3)F),&F,F3F3F-F-,&F3F-F0F-F3,&F3F-F)F-F3" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXU No q-hypergeometric antidifference \+ exists.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "qgosper(term(2 ,k),q,k,simplify=false);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*0,&\"\"\" !\"\")%\"qG%\"kGF%F%%\"aGF%F(\"\"#,(*&F%F%,&F%F&*&F*F%)F(F+F%F%F&F%*.F .F&,&F%F&*&F*F%F(F%F%F&F*F%,&*$F0F%F%F%F&F%F'F%F(F&F%*2F*F+,&F(F%F%F&F %,&F*F%F%F&F&F.F&F2F&F4F%F'F+F(!\"#F%F%-%,qpochhammerG6%F*F(F)F%-F;6%F (F(F)F&)F/F)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "qgosper(t erm(n,k),q,k,series=true);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*0,&!\" \"\"\"\")%\"qG%\"kGF&F&%\"aGF&)F(%\"nGF&-%$SumG6$**-%,qpochhammerG6%)F (,$F,F%F(%#_iGF&,&*&F*F&F+F&F&)F(F6F%F%-F26%*&F*F%F4F&F(F6F%)F(*&F)F&F 6F&F&/F6;\"\"!%)infinityGF&-F26%F*F(F)F&-F26%F(F(F)F%)F8,$F)F%F&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Note that the above sum is not rea lly infinite if " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 33 " is a pos itive integer. Thus the " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 90 "- hypergeometric antidifference returned by qgosper contains aVF76$F=F*F( F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "term:= (n,k) -> qpoch hammer(a,q,k)/qpochhammer(q,q,k)/(a*q^n)^k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%termGf*6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF)*(-%,qp ochhammerG6%%\"aG%\"qG9%\"\"\"-F/6%F2F2F3!\"\")*&F1F4)F29$F4F3F7F)F)F) " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "This is an example where qgos per decides that there is no " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 45 "-hypergeometric antidifference for arbitrary " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 37 ". However, for each positive integer " } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 19 " qgosper returns a " } {XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 48 "-hypergeometric antidifferen ce. With the option " }{TEXT 19 11 "series=true" }{TEXT -1 60 ", we ar e even able to prove that for every positive integer " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 31 " such an antidifference exists." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "qgosper(term(n,k),q,k);" }}{PARA 8 "" 1 " " {TEXT -1 63 "Error, (in qgosper)WPMATH 20 "6#**,&\"\"\"!\"\"*&%\"aGF%)% \"qG%\"kGF%F%F%,&F(F%F%F&F&-%,qpochhammerG6%F(F*F+F%-F.6%F*F*F+F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "qgosper(qpochhammer(a,q,k)* qpochhammer(a*q^2,q^2,k)*\n qpochhammer(q^(-n),q,k)*q^(n*k)/(qp ochhammer(a,q^2,k)*\n qpochhammer(a*q^(n+1),q,k)*qpochhammer(q, q,k)),q,k);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*8,&\"\"\"!\"\"*&% \"aGF&)%\"qG,&%\"kGF&%\"nGF&F&F&F&,&F&F')F+F-F&F&,&)F+F.F&F&F'F',&F&F' *&F)F&)F+,$*&\"\"#F&F-F&F&F&F&F'-%,qpochhammerG6%F)F+F-F&-F:6%*&F)F&)F +F8F&*$F?F&F-F&-F:6%)F+,$F.F'F+F-F&)F+*&F.F&F-F&F&-F:6%F)F@F-F'-F:6%*& F)F&)F+,&F.F&F&F&F&F+F-F'-F:6%F+F+F-F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "qgosper(q^k*qbrackets(k,q),q,k=m..n,simplify=false); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,& **,(*(%$_C1G\"\"\")%\"qG,&%\"nGF(F(F(!\"\"F*F(F(*&F(F(,&F*F(F(F-F-F-*& ,&*$)F*\"\"#F(F(F(F-F-F)F(F(F(,&F(F-F)F(F-F)F(-%*qbracketsG6$F+F*F(F(* *,(*(F'F()F*%\"mGF-F*F(F(F.F-*&F1F-F " 0 "" {MPLTEXT 1 0 17 "rea d `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%PPackage~\"q-Hyperg eometric~Summation\",~Maple~V-8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%c oCopyright~1998-2002,~Harald~Boeing~&~Wolfram~Koepf,~University~of~Kas selG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "qgosper(qpochhammer (a,q,k)*q^k/qpochhammer(q,q,k),q,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#**,&\"\"\"!\"\")%\"qG%\"kGF%F%,&%\"aGF%F%F&F&-%,qpochhammerG6%F+F(F) F%-F-6%F(F(F)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "qgosper( qpochhammer(a,q,k)*q^k/qpochhammer(q,q,k),q,k,\n antidifference =down);" }}{PARA 11 "" 1 "" {XPY"6#\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" }{TEXT 256 2 " " }{HYPERLNK 17 "qbrackets, qbinomial, q factorial, qGAMMA & qpochhammer" 2 "qfunctions" "" }{TEXT 260 2 ", " } {HYPERLNK 17 "qsimpcomb" 2 "qsimpcomb" "" }{TEXT 261 9 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ummation\",~Maple~V-8G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boeing~&~Wolf ram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "f:= qpochhammer(a*q^(-k*n),q,n)-qpochhammer(q/a,q,k* n)/\nqpochhammer(q/a,q,k*n-n)*(-a)^n*q^(binomial(n,2)-k*n^2);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"fG,&-%,qpochhammerG6%*&%\"aG\"\"\" )%\"qG,$*&%\"kGF+%\"nGF+!\"\"F+F-F1F+**-F'6%*&F-F+F*F2F-F/F+-F'6%F6F-, &F/F+F1F2F2),$F*F2F1F+)F-,&-%)binomialG6$F1\"\"#F+*&F0F+)F1FAF+F2F+F2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qsimplify(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 \$ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpeadingnormalheadbulletitemfunctqhypertermproducqghypergeometrictermqpsihypertermcallsequencnqphihypertermnqpsihpertermparameterlistalgebraicexpressnqnameintegnzheevaluatpointnkdescriptfunctionsprovidpossibilitentersummandbasichypergeometricseribilateralinvolvowerqbracketqbinomialqfactorialqpochhammerqfunctphiphiguglgzgdefinsumproductqpochhammproductbinomialinfinitsumgproductgmmergigfkgrgfsgfffbinomialgfefffinfinitygnwherwithrgentrrespectivereturnqphihypertermaliapsipsigqpochhammqpochhammerginfinitygfffundexamplreadqsummplppackagsummatcocopyrightharaldboeingwolframkoepfuniversitkasselgagbgfzgfcgfgfalsoqsimpcombqsumrecursPERLNK 17 "qsumrecursion" 2 "qsumrecursion" "" }{TEXT 266 6 " \+ " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } PARA 11 "" 1 "" {XPPMATH 20 "6#%PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boeing ~&~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "qphihyperterm([a,b],[c],q,z,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,-%,qpochhammerG6%%\"aG%\"qG%\"kG\"\"\"-F%6%%\"bGF(F)F *)%\"zGF)F*-F%6%%\"cGF(F)!\"\"-F%6%F(F(F)F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "qpsihyperterm([a,b],[c],q,z,k);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*0-%,qpochhammerG6%%\"aG%\"qG%\"kG\"\"\"-F%6%%\"bGF(F )F*)!\"\",$F)F/F*)F(,$*(\"\"#F/F)F*,&F)F*F*F/F*F/F*)%\"zGF)F*-F%6%%\"c GF(F)F/-F%6%F(F(F)F/" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also :" }{TEXT 256 2 " " }{HYPERLNK 17 "qsimpcomb" 2 "qsimpcomb" "" } {TEXT 258 2 ", " }{HYPERLNK 17 "qsum" 2 "qsum" "" }{TEXT 259 2 ", " } {HY^lateral basic hypergeometric serie s." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boeing ~&~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "qphihyperterm([a,b],[c],q,z,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,-%,qpochhammerG6%%\"aG%\"qG%\"kG\"\"\"-F%6%%\"bGF(F)F *)%\"zGF)F*-F%6%%\"cGF(F)!\"\"-F%6%F(F(F)F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "qpsihyperterm([a,b],[c],q,z,k);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*0-%,qpochhammerG6%%\"aG%\"qG%\"kG\"\"\"-F%6%%\"bGF(F )F*)!\"\",$F)F/F*)F(,$*(\"\"#F/F)F*,&F)F*F*F/F*F/F*)%\"zGF)F*-F%6%%\"c GF(F)F/-F%6%F(F(F)F/" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also :" }{TEXT 256 2 " " }{HYPERLNK 17 "qsimpcomb" 2 "qsimpcomb" "" } {TEXT 258 2 ", " }{HYPERLNK 17 "qsum" 2 "qsum" "" }{TEXT 259 2 ", " } {HY^q,z, k)" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 264 39 "bilateral basic hypergeometric function" }{TEXT -1 1 " " } {XPPEDIT 18 0 "psi(U,L,q,z)" "6#-%$psiG6&%\"UG%\"LG%\"qG%\"zG" }{TEXT -1 16 " is defined as:\n" }{XPPEDIT 18 0 "psi(U,L,q,z)=sum(product(qpo chhammer(U[i],q,k),i=1..r)/product(qpochhammer(L[i],q,k),i=1..s)*z^k*( (-1)^k*q^binomial(k,2))^(s-r),k=-infinity..infinity)" "6#/-%$psiG6&%\" UG%\"LG%\"qG%\"zG-%$sumG6$**-%(productG6$-%,qpochhammerG6%&F'6#%\"iGF) %\"kG/F7;\"\"\"%\"rGF;-F06$-F36%&F(6#F7F)F8/F7;F;%\"sG!\"\")F*F8F;)*&) ,$F;FFF8F;)F)-%)binomialG6$F8\"\"#F;,&FEF;F and Koepf, Wolfram " }}{PARA 15 "" 0 "" {TEXT 260 32 "Topics: Analysis, Combinato rics" }}}{SECT 0 {PARA 4 "" 0 "examples" {TEXT 261 9 "Examples:" }} {PARA 15 "" 0 "" {TEXT 262 146 "To load this package, type the followi ng Maple commands in a Maple session (or cut and paste them, if you ar e us$*:\"\"#\"\" \")!\"\"%\"jGF+-%*factorialG6#,$*&F*F+F.F+F+F+,&F+F+*&F*F+F.F+F+F+-F06 #F.!\"#,&F.F+F+F+F-,&F.F+F*F+F-)%\"xGF.F+F " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "seealso" {TEXT -1 10 "See Also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "FPS[FormalPowerSeries]" 2 "FPS[FormalPowerSeries]" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "FPS[HolonomicDE]," 2 "HolonomicDE," "" }{TEXT -1 1 " " }{HYPERLNK 17 "FPS[SimpleRE]." 2 "FPS[SimpleRE]" "" }} }}{MARK "6 1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ATH 20 "6#>%$fpsG-%$Su mG6$,$*2\"\"#\"\"\")!\"\"%\"jGF+-%*factorialG6#,&F+F+*&F*F+F.F+F+F+-F0 6#F.!\"#,&F.F+F+F+F-,&F.F+F*F+F-)%\"xGF7F+-F&6$*(-%+pochhammerG6$,&*&F *F+F.F+F+F*F+%\"kGF+-F06#FCF-)%#z|irG,(FCF+F+F+F.F+F+/FC;\"\"!%)infini tyGF+F+/F.FJ" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "standardsum (fps);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$-F$6$,hhed helpM helvetica henc heor herehg hhammhighhodholonom holonomic  holonomicd;@    homo  homogeneo homogeneou_ inhomogeneou'inifcn inikinit initialinityg inomialinputXins instinsteadinstitut int?intdiff intdiffeqG,algorialgorith algorithm{R algorithmicaliaalls*allow' allyalmkvistalogu alpalpha+alphag {VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 25 "Courier" 0 1 0 0 0 1 2 2 0 0 0 0 0 0 0 1 }{CSTYLE "Help Norma l" -1 30 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 " courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" 23 259 "courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Hea ding 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 AW{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "times" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "times" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "times" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268$-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qsimplify(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" }{TEXT 256 1 " " } {HYPERLNK 17 "qbrackets,qpochhammer,qfactorial, qGAMMA and qbinomial" 2 "qfunctions" "" }{TEXT 260 2 ", " }{HYPERLNK 17 "qhyperterm and qpsi hyperterm" 2 "qhyperterm" "" }{TEXT 259 2 ", " }{HYPERLNK 17 "qsimplif y" 2 "qsimplify" "" }{TEXT 261 2 ", " }{HYPERLNK 17 "qsum" 2 "qsum" " " }{TEXT 258 8 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ochha mmer(a,q^2,n)*\nqpochhammer(a*q,q^2,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(-%,qpochhammerG6%%\"aG%\"qG,$*&\"\"#\"\"\"%\"nGF.F.!\"\" -F'6%F)*$)F*F-F.F/F.-F'6%*&F)F.F*F.F3F/F." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "qsimpcomb(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*( -%,qpochhammerG6%%\"aG%\"qG,$*&\"\"#\"\"\"%\"nGF,F,!\"\"-F%6%F'*$)F(F+ F,F-F,-F%6%*&F'F,F(F,F1Fo**-% ,qpochhammerG6%%\"aG%\"qG,$%\"nG!\"\"F-),$*&F*\"\"\"F)F-F-F,F1)F*-%)bi nomialG6$F,\"\"#F1-F'6%F0F*F,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "qsimpcomb(subs(n=n+1,f)/f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "qsimpcomb(qpochh ammer(a*q,q,k+1)/qpochhammer(a,q,k-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,&\"\"\"!\"\"*&%\"aGF%)%\"qG%\"kGF%F%F%,&F%F&*(F(F%F)F%F*F%F%F %,&F*F&F'F%F%,&F%F&F(F%F&F*F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 " Note: f is equal to one, but qsimpcomb fails to simplify this term." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f:= 1/qpochhammer(a,q,2*n)*qpochha mmer(a,q^2,n)*\nqpochhammer(a*q,q^2,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(-%,qpochhammerG6%%\"aG%\"qG,$*&\"\"#\"\"\"%\"nGF.F.!\"\" -F'6%F)*$)F*F-F.F/F.-F'6%*&F)F.F*F.F3F/F." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "qsimpcomb(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*( -%,qpochhammerG6%%\"aG%\"qG,$*&\"\"#\"\"\"%\"nGF,F,!\"\"-F%6%F'*$)F(F+ F,F-F,-F%6%*&F'F,F(F,F1FLTEXT 1 0 26 "qsimpcomb(qbrackets( k,q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&)%\"qG%\"kG\"\"\"F(!\"\" F(,&F&F(F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "qsimpcomb (qbinomial(n,2,q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,,&%\"qG\"\" \")F&%\"nG!\"\"F',&F'F*F(F'F',&F&F'F'F*!\"#,&F&F'F'F'F*F&F*F*" }}}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" }{TEXT -1 1 " " } {HYPERLNK 17 "qhyperterm&qpsihyperterm" 2 "qhyperterm" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qsimpcomb" 2 "qsimpcomb" "" }{TEXT -1 3 " " }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ~Boeing~&~Wolfram~Koep f,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The p rocedure qsimpcomb provides a simplification mechanism for all those f unctions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "qsimpcomb(qpochhammer( a,q,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&\"\"\"!\"\"%\"aGF%F%,& F%F&*&F'F%%\"qGF%F%F%,&F%F&*&F'F%)F*\"\"#F%F%F%,&F%F&*&F'F%)F*\"\"$F%F %F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPsLTEXT 1 0 26 "qsimpcomb(qbrackets( k,q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&)%\"qG%\"kG\"\"\"F(!\"\" F(,&F&F(F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "qsimpcomb (qbinomial(n,2,q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,,&%\"qG\"\" \")F&%\"nG!\"\"F',&F'F*F(F'F',&F&F'F'F*!\"#,&F&F'F'F'F*F&F*F*" }}}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" }{TEXT -1 1 " " } {HYPERLNK 17 "qhyperterm&qpsihyperterm" 2 "qhyperterm" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qsimpcomb" 2 "qsimpcomb" "" }{TEXT -1 3 " " }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ~Boeing~&~Wolfram~Koep f,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The p rocedure qsimpcomb provides a simplification mechanism for all those f unctions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "qsimpcomb(qpochhammer( a,q,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&\"\"\"!\"\"%\"aGF%F%,& F%F&*&F'F%%\"qGF%F%F%,&F%F&*&F'F%)F*\"\"#F%F%F%,&F%F&*&F'F%)F*\"\"$F%F %F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPuLTEXT 1 0 26 "qsimpcomb(qbrackets( k,q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&)%\"qG%\"kG\"\"\"F(!\"\" F(,&F&F(F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "qsimpcomb (qbinomial(n,2,q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,,&%\"qG\"\" \")F&%\"nG!\"\"F',&F'F*F(F'F',&F&F'F'F*!\"#,&F&F'F'F'F*F&F*F*" }}}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" }{TEXT -1 1 " " } {HYPERLNK 17 "qhyperterm&qpsihyperterm" 2 "qhyperterm" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qsimpcomb" 2 "qsimpcomb" "" }{TEXT -1 3 " " }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ~Boeing~&~Wolfram~Koep f,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The p rocedure qsimpcomb provides a simplification mechanism for all those f unctions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "qsimpcomb(qpochhammer( a,q,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&\"\"\"!\"\"%\"aGF%F%,& F%F&*&F'F%%\"qGF%F%F%,&F%F&*&F'F%)F*\"\"#F%F%F%,&F%F&*&F'F%)F*\"\"$F%F %F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPwLTEXT 1 0 26 "qsimpcomb(qbrackets( k,q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&)%\"qG%\"kG\"\"\"F(!\"\" F(,&F&F(F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "qsimpcomb (qbinomial(n,2,q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,,&%\"qG\"\" \")F&%\"nG!\"\"F',&F'F*F(F'F',&F&F'F'F*!\"#,&F&F'F'F'F*F&F*F*" }}}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" }{TEXT -1 1 " " } {HYPERLNK 17 "qhyperterm&qpsihyperterm" 2 "qhyperterm" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qsimpcomb" 2 "qsimpcomb" "" }{TEXT -1 3 " " }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ~Boeing~&~Wolfram~Koep f,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The p rocedure qsimpcomb provides a simplification mechanism for all those f unctions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "qsimpcomb(qpochhammer( a,q,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&\"\"\"!\"\"%\"aGF%F%,& F%F&*&F'F%%\"qGF%F%F%,&F%F&*&F'F%)F*\"\"#F%F%F%,&F%F&*&F'F%)F*\"\"$F%F %F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPyLTEXT 1 0 26 "qsimpcomb(qbrackets( k,q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&)%\"qG%\"kG\"\"\"F(!\"\" F(,&F&F(F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "qsimpcomb (qbinomial(n,2,q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,,&%\"qG\"\" \")F&%\"nG!\"\"F',&F'F*F(F'F',&F&F'F'F*!\"#,&F&F'F'F'F*F&F*F*" }}}} {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" }{TEXT -1 1 " " } {HYPERLNK 17 "qhyperterm&qpsihyperterm" 2 "qhyperterm" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "qsimpcomb" 2 "qsimpcomb" "" }{TEXT -1 3 " " }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ~Boeing~&~Wolfram~Koep f,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The p rocedure qsimpcomb provides a simplification mechanism for all those f unctions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "qsimpcomb(qpochhammer( a,q,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&\"\"\"!\"\"%\"aGF%F%,& F%F&*&F'F%%\"qGF%F%F%,&F%F&*&F'F%)F*\"\"#F%F%F%,&F%F&*&F'F%)F*\"\"$F%F %F%" }}}{EXCHG {PARA 0 "> " 0 "" {MP{@{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 2ents which both represent any valid algebraic expression. It is defined as" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "qGAMMA(z,q)=qpochhamme r(q,q,infinity)/qpochhammer(q^z,q,infinity)*(1-q)^(1-z)" "6#/-%'qGAMMA G6$%\"zG%\"qG*(-%,qpochhammerG6%F(F(%)infinityG\"\"\"-F+6%)F(F'F(F-!\" \"),&F.F.F(F2,&F.F.F'F2F." }{TEXT -1 1 "." }}}}{SECT 0 {PARA 0 "" 0 " " {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boeing~&~Wolfram~Koep f,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The p rocedure qsimpcomb provides a simplification mechanism for all those f unctions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "qsimpcomb(qpochhammer( a,q,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&\"\"\"!\"\"%\"aGF%F%,& F%F&*&F'F%%\"qGF%F%F%,&F%F&*&F'F%)F*\"\"#F%F%F%,&F%F&*&F'F%)F*\"\"$F%F %F%" }}}{EXCHG {PARA 0 "> " 0 "" {MP{nomial function uses three arguments, where the first two are any valid algebraic expressi ons representing integer values (symbolic) and the last one is a name \+ or a name raised to an integer power. The function is defined as:" }} {PARA 259 "" 0 "" {XPPEDIT 18 0 "qbinomial(n,k,q) = qpochhammer(q,q,n) /(qpochhammer(q,q,k)*qpochhammer(q,q,n-k)) " "6#/-%*qbinomialG6%%\"nG% \"kG%\"qG*&-%,qpochhammerG6%F)F)F'\"\"\"*&-F,6%F)F)F(F.-F,6%F)F),&F'F. F(!\"\"F.F5" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "qfactorial:" } {TEXT -1 222 " The qfactorial function has two arguments. The first on e is any valid algebraic expression that represents an integer value ( symbolic) and the second one is a name or a name raised to any integer power. It is defined as: " }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "qfactor ial(k,q) = qpochhammer(q,q,k)/(1-q)^k" "6#/-%+qfactorialG6$%\"kG%\"qG* &-%,qpochhammerG6%F(F(F'\"\"\"),&F-F-F(!\"\"F'F0" }}}{SECT 0 {PARA 0 " " 0 "" {TEXT 26 7 "qGAMMA:" }{TEXT -1 108 " The qGAMMA function has tw o argum}alphagf alreadalso [alway#amame americal ammerammerganalanaloanalogonanalogu#analysi andardsumanishannotanothantiantid antideriv antiderivat antidifantidiff antidifferen antidifferenc*  antidifferencegany{Gapleappear appendixappl+appli applicabl applicat?apply approach1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List It em" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "" 3 256 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 3 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 14 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 2 " \+ " }{TEXT 256 10 "qgosper - " }{TEXT 265 1 "q" }{TEXT 266 31 "-analogue of Gosper's algorithm" }}{PARA 4 "" 0 "" {TEXT 26 18 "Calling Sequenc es:" }{TEXT 263 1 " " }}{PARA 256 "" 0 "" {TEXT -1 222 "qgosper(f,q,k, ...);\nqgauto availabavailablavoidbasebased baselg#basic/bateman behabehaviorbelowbergbesselibeta betag bettbgO0bgf;bi bibliograph bilateral binbinoconsiderconsistconstant consumconta contactcontain/context contgocontgosp/ continuou controcontrolconvers convertcopi copyrightcorrect correspod correspond/reg relatreleasremainremembrencrentialrepreport repres+ represent# representatreqrequir res resentatresprespect+ respective3restrictresulresult[!expressi expressioextend extensextent fa fac factor  factorial' factorialg?4factoriz factorizatfacult faffail# falsfamilifasenfasenmy3outputoutsidover#ovsek owerpa packpackag+paramparamet; parameter3part759 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help for:" }{TEXT -1 7 " B asic " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 10 "-functions" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 110 "qpochhammer(a,q,infinity);\nqpochhammer(a,q, k);\nqfac(a,q,k);\nqbrackets(k,q);\nqfactorial(k,q);\nqbinomial(n,k,q) ;" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 13 "Description: " }}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "qpochhammer:" }{TEXT 2 284 " The qpochham mer function takes three arguments. The first parameter is any algebra ic expression, the second one is of type name or name raised ot an int eger power and the third variable is an algebraic expression represent ing an i(ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalasicqgfunctioncallsequencqpochhamminfinitnqpochhammnqfacnqbracketnqfactorialnqbinomialdescriptqpochhammerfuncttakeargumentfirstparametanyalgebraicexpresssecondtypenameraisotintegerpowerthirdvariablalgebraicrepresingintegvalusymbolicdefinproductqpochhammergaginfinitygproductgjgfergkgpiecewisegproductgqfacunctsynonymqbracketbothcanvaliddefinitqbracketsgqbinomialusesexpressionsrepresentlastqbinomialgngqfactorialqfactorialqfactorialgqgammatwzgexamplreadqsummplppackaghypergeometricsummatcocopyrightharaldboeingwolframkoepuniversitkasselgrocedurqsimpcombprovidsimplificatmechanismallthosunctionagfqgfalsoqhypertermqpsihyperterm@{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 2ents which both represent any valid algebraic expression. It is defined as" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "qGAMMA(z,q)=qpochhamme r(q,q,infinity)/qpochhammer(q^z,q,infinity)*(1-q)^(1-z)" "6#/-%'qGAMMA G6$%\"zG%\"qG*(-%,qpochhammerG6%F(F(%)infinityG\"\"\"-F+6%)F(F'F(F-!\" \"),&F.F.F(F2,&F.F.F'F2F." }{TEXT -1 1 "." }}}}{SECT 0 {PARA 0 "" 0 " " {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boeing~&~Wolfram~Koep f,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The p rocedure qsimpcomb provides a simplification mechanism for all those f unctions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "qsimpcomb(qpochhammer( a,q,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&\"\"\"!\"\"%\"aGF%F%,& F%F&*&F'F%%\"qGF%F%F%,&F%F&*&F'F%)F*\"\"#F%F%F%,&F%F&*&F'F%)F*\"\"$F%F %F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPynomial function uses three arguments, where the first two are any valid algebraic expressi ons representing integer values (symbolic) and the last one is a name \+ or a name raised to an integer power. The function is defined as:" }} {PARA 259 "" 0 "" {XPPEDIT 18 0 "qbinomial(n,k,q) = qpochhammer(q,q,n) /(qpochhammer(q,q,k)*qpochhammer(q,q,n-k)) " "6#/-%*qbinomialG6%%\"nG% \"kG%\"qG*&-%,qpochhammerG6%F)F)F'\"\"\"*&-F,6%F)F)F(F.-F,6%F)F),&F'F. F(!\"\"F.F5" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "qfactorial:" } {TEXT -1 222 " The qfactorial function has two arguments. The first on e is any valid algebraic expression that represents an integer value ( symbolic) and the second one is a name or a name raised to any integer power. It is defined as: " }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "qfactor ial(k,q) = qpochhammer(q,q,k)/(1-q)^k" "6#/-%+qfactorialG6$%\"kG%\"qG* &-%,qpochhammerG6%F(F(F'\"\"\"),&F-F-F(!\"\"F'F0" }}}{SECT 0 {PARA 0 " " 0 "" {TEXT 26 7 "qGAMMA:" }{TEXT -1 108 " The qGAMMA function has tw o argumnteger value (symbolic). The function is defined as: " }} {PARA 257 "" 0 "" {XPPEDIT 18 0 "qpochhammer(a,q,infinity)=product(1-a *q^j,j=0..infinity)" "6#/-%,qpochhammerG6%%\"aG%\"qG%)infinityG-%(prod uctG6$,&\"\"\"F.*&F'F.)F(%\"jGF.!\"\"/F1;\"\"!F)" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 2 "6#/-%,qpochhamm erG6%%\"aG%\"qG%\"kG-%*PIECEWISEG6%7$-%(productG6$,&\"\"\"F2*&F'F2)F(% \"jGF2!\"\"/F5;\"\"!,&F)F2F6F22F9F)7$F2/F)F97$-F/6$*$F1F6/F5;F)F22F)F9 " }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 5 "qfac:" }{TEXT -1 81 " This f unction is a synonym for qpochhammer, i.e. qfac(a,q,k)=qpochhammer(a,q ,k)." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "qbrackets:" }{TEXT -1 105 " The qbrackets function has two arguments. Both can be any valid \+ algebraic expression. The definition is:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "qbrackets(k,q) = (q^k-1)/(q-1)" "6#/-%*qbracketsG6$%\"k G%\"qG*&,&)F(F'\"\"\"F,!\"\"F,,&F(F,F,F-F-" }}}{SECT 0 {PARA 0 "" 0 " " {TEXT 26 10 "qbinomial:" }{TEXT -1 233 " The qbi59 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help for:" }{TEXT -1 7 " B asic " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 10 "-functions" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 110 "qpochhammer(a,q,infinity);\nqpochhammer(a,q, k);\nqfac(a,q,k);\nqbrackets(k,q);\nqfactorial(k,q);\nqbinomial(n,k,q) ;" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 13 "Description: " }}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "qpochhammer:" }{TEXT 2 284 " The qpochham mer function takes three arguments. The first parameter is any algebra ic expression, the second one is of type name or name raised ot an int eger power and the third variable is an algebraic expression represent ing an icannotcascase;catececedur certain certicertifcertifi certifica certificat3cfcg#cgf;chchanc changCReferences for the packag e hsum:" }}{EXCHG {PARA 15 "" 0 "" {TEXT -1 112 "Almkvist, G. and Zeil berger, D.: The method of differentiating under the integral sign. J. \+ Symbolic Computation " }{TEXT 256 2 "10" }{TEXT -1 15 ", 1990, 571-591 " }}{PARA 15 "" 0 "" {TEXT -1 105 "Gosper Jr., R. W.: Decision procedu re for indefinite hypergeometric summation. Proc. Nat. Acad. Sci. USA \+ " }{TEXT 257 2 "75" }{TEXT -1 15 " (1978), 40-42." }}{PARA 15 "" 0 "" {TEXT -1 26 "Koepf, W.: Algorithms for " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 64 "-fold hypergeometric summation. Journal of Symbolic Comp utation " }{TEXT 258 2 "20" }{TEXT -1 16 ", 1995, 399-417." }}{PARA 15 "" 0 "" {TEXT -1 164 "Koepf, W.: Hypergeometric Summation, An Algor ithmic Approach to Summation and Special Function Identities. Vieweg, \+ Braunschweig/Wiesbaden, 1998, ISBN 3-528-06950-3." }}{PARA 15 "" 0 "" {TEXT -1 54 "Koornwinder, T. H.: On Zeilberger's algorithm and its " } {TEXT 261 1 "q" }{TEXT -1 65 "-analogue: a rigorous description. J.roductrota routinrovidenc rrencrrorrsionrstrtcut rtificat ruerulerun sagesamesatisfsatisfisaving schem sci@{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 2ents which both represent any valid algebraic expression. It is defined as" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "qGAMMA(z,q)=qpochhamme r(q,q,infinity)/qpochhammer(q^z,q,infinity)*(1-q)^(1-z)" "6#/-%'qGAMMA G6$%\"zG%\"qG*(-%,qpochhammerG6%F(F(%)infinityG\"\"\"-F+6%)F(F'F(F-!\" \"),&F.F.F(F2,&F.F.F'F2F." }{TEXT -1 1 "." }}}}{SECT 0 {PARA 0 "" 0 " " {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boeing~&~Wolfram~Koep f,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The p rocedure qsimpcomb provides a simplification mechanism for all those f unctions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "qsimpcomb(qpochhammer( a,q,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&\"\"\"!\"\"%\"aGF%F%,& F%F&*&F'F%%\"qGF%F%F%,&F%F&*&F'F%)F*\"\"#F%F%F%,&F%F&*&F'F%)F*\"\"$F%F %F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPwnomial function uses three arguments, where the first two are any valid algebraic expressi ons representing integer values (symbolic) and the last one is a name \+ or a name raised to an integer power. The function is defined as:" }} {PARA 259 "" 0 "" {XPPEDIT 18 0 "qbinomial(n,k,q) = qpochhammer(q,q,n) /(qpochhammer(q,q,k)*qpochhammer(q,q,n-k)) " "6#/-%*qbinomialG6%%\"nG% \"kG%\"qG*&-%,qpochhammerG6%F)F)F'\"\"\"*&-F,6%F)F)F(F.-F,6%F)F),&F'F. F(!\"\"F.F5" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "qfactorial:" } {TEXT -1 222 " The qfactorial function has two arguments. The first on e is any valid algebraic expression that represents an integer value ( symbolic) and the second one is a name or a name raised to any integer power. It is defined as: " }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "qfactor ial(k,q) = qpochhammer(q,q,k)/(1-q)^k" "6#/-%+qfactorialG6$%\"kG%\"qG* &-%,qpochhammerG6%F(F(F'\"\"\"),&F-F-F(!\"\"F'F0" }}}{SECT 0 {PARA 0 " " 0 "" {TEXT 26 7 "qGAMMA:" }{TEXT -1 108 " The qGAMMA function has tw o argumnteger value (symbolic). The function is defined as: " }} {PARA 257 "" 0 "" {XPPEDIT 18 0 "qpochhammer(a,q,infinity)=product(1-a *q^j,j=0..infinity)" "6#/-%,qpochhammerG6%%\"aG%\"qG%)infinityG-%(prod uctG6$,&\"\"\"F.*&F'F.)F(%\"jGF.!\"\"/F1;\"\"!F)" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 2 "6#/-%,qpochhamm erG6%%\"aG%\"qG%\"kG-%*PIECEWISEG6%7$-%(productG6$,&\"\"\"F2*&F'F2)F(% \"jGF2!\"\"/F5;\"\"!,&F)F2F6F22F9F)7$F2/F)F97$-F/6$*$F1F6/F5;F)F22F)F9 " }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 5 "qfac:" }{TEXT -1 81 " This f unction is a synonym for qpochhammer, i.e. qfac(a,q,k)=qpochhammer(a,q ,k)." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "qbrackets:" }{TEXT -1 105 " The qbrackets function has two arguments. Both can be any valid \+ algebraic expression. The definition is:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "qbrackets(k,q) = (q^k-1)/(q-1)" "6#/-%*qbracketsG6$%\"k G%\"qG*&,&)F(F'\"\"\"F,!\"\"F,,&F(F,F,F-F-" }}}{SECT 0 {PARA 0 "" 0 " " {TEXT 26 10 "qbinomial:" }{TEXT -1 233 " The qbi59 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help for:" }{TEXT -1 7 " B asic " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 10 "-functions" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 110 "qpochhammer(a,q,infinity);\nqpochhammer(a,q, k);\nqfac(a,q,k);\nqbrackets(k,q);\nqfactorial(k,q);\nqbinomial(n,k,q) ;" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 13 "Description: " }}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "qpochhammer:" }{TEXT 2 284 " The qpochham mer function takes three arguments. The first parameter is any algebra ic expression, the second one is of type name or name raised ot an int eger power and the third variable is an algebraic expression represent ing an i(ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalasicqgfunctioncallsequencqpochhamminfinitnqpochhammnqfacnqbracketnqfactorialnqbinomialdescriptqpochhammerfuncttakeargumentfirstparametanyalgebraicexpresssecondtypenameraisotintegerpowerthirdvariablalgebraicrepresingintegvalusymbolicdefinproductqpochhammergaginfinitygproductgjgfergkgpiecewisegproductgqfacunctsynonymqbracketbothcanvaliddefinitqbracketsgqbinomialusesexpressionsrepresentlastqbinomialgngqfactorialqfactorialqfactorialgqgammatwzgexamplreadqsummplppackaghypergeometricsummatcocopyrightharaldboeingwolframkoepuniversitkasselgrocedurqsimpcombprovidsimplificatmechanismallthosunctionagfqgfalsoqhypertermqpsihyperterm@{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 2ents which both represent any valid algebraic expression. It is defined as" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "qGAMMA(z,q)=qpochhamme r(q,q,infinity)/qpochhammer(q^z,q,infinity)*(1-q)^(1-z)" "6#/-%'qGAMMA G6$%\"zG%\"qG*(-%,qpochhammerG6%F(F(%)infinityG\"\"\"-F+6%)F(F'F(F-!\" \"),&F.F.F(F2,&F.F.F'F2F." }{TEXT -1 1 "." }}}}{SECT 0 {PARA 0 "" 0 " " {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boeing~&~Wolfram~Koep f,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The p rocedure qsimpcomb provides a simplification mechanism for all those f unctions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "qsimpcomb(qpochhammer( a,q,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&\"\"\"!\"\"%\"aGF%F%,& F%F&*&F'F%%\"qGF%F%F%,&F%F&*&F'F%)F*\"\"#F%F%F%,&F%F&*&F'F%)F*\"\"$F%F %F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPunomial function uses three arguments, where the first two are any valid algebraic expressi ons representing integer values (symbolic) and the last one is a name \+ or a name raised to an integer power. The function is defined as:" }} {PARA 259 "" 0 "" {XPPEDIT 18 0 "qbinomial(n,k,q) = qpochhammer(q,q,n) /(qpochhammer(q,q,k)*qpochhammer(q,q,n-k)) " "6#/-%*qbinomialG6%%\"nG% \"kG%\"qG*&-%,qpochhammerG6%F)F)F'\"\"\"*&-F,6%F)F)F(F.-F,6%F)F),&F'F. F(!\"\"F.F5" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "qfactorial:" } {TEXT -1 222 " The qfactorial function has two arguments. The first on e is any valid algebraic expression that represents an integer value ( symbolic) and the second one is a name or a name raised to any integer power. It is defined as: " }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "qfactor ial(k,q) = qpochhammer(q,q,k)/(1-q)^k" "6#/-%+qfactorialG6$%\"kG%\"qG* &-%,qpochhammerG6%F(F(F'\"\"\"),&F-F-F(!\"\"F'F0" }}}{SECT 0 {PARA 0 " " 0 "" {TEXT 26 7 "qGAMMA:" }{TEXT -1 108 " The qGAMMA function has tw o argumnteger value (symbolic). The function is defined as: " }} {PARA 257 "" 0 "" {XPPEDIT 18 0 "qpochhammer(a,q,infinity)=product(1-a *q^j,j=0..infinity)" "6#/-%,qpochhammerG6%%\"aG%\"qG%)infinityG-%(prod uctG6$,&\"\"\"F.*&F'F.)F(%\"jGF.!\"\"/F1;\"\"!F)" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 2 "6#/-%,qpochhamm erG6%%\"aG%\"qG%\"kG-%*PIECEWISEG6%7$-%(productG6$,&\"\"\"F2*&F'F2)F(% \"jGF2!\"\"/F5;\"\"!,&F)F2F6F22F9F)7$F2/F)F97$-F/6$*$F1F6/F5;F)F22F)F9 " }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 5 "qfac:" }{TEXT -1 81 " This f unction is a synonym for qpochhammer, i.e. qfac(a,q,k)=qpochhammer(a,q ,k)." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "qbrackets:" }{TEXT -1 105 " The qbrackets function has two arguments. Both can be any valid \+ algebraic expression. The definition is:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "qbrackets(k,q) = (q^k-1)/(q-1)" "6#/-%*qbracketsG6$%\"k G%\"qG*&,&)F(F'\"\"\"F,!\"\"F,,&F(F,F,F-F-" }}}{SECT 0 {PARA 0 "" 0 " " {TEXT 26 10 "qbinomial:" }{TEXT -1 233 " The qbi59 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help for:" }{TEXT -1 7 " B asic " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 10 "-functions" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 110 "qpochhammer(a,q,infinity);\nqpochhammer(a,q, k);\nqfac(a,q,k);\nqbrackets(k,q);\nqfactorial(k,q);\nqbinomial(n,k,q) ;" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 13 "Description: " }}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "qpochhammer:" }{TEXT 2 284 " The qpochham mer function takes three arguments. The first parameter is any algebra ic expression, the second one is of type name or name raised ot an int eger power and the third variable is an algebraic expression represent ing an i(ibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalasicqgfunctioncallsequencqpochhamminfinitnqpochhammnqfacnqbracketnqfactorialnqbinomialdescriptqpochhammerfuncttakeargumentfirstparametanyalgebraicexpresssecondtypenameraisotintegerpowerthirdvariablalgebraicrepresingintegvalusymbolicdefinproductqpochhammergaginfinitygproductgjgfergkgpiecewisegproductgqfacunctsynonymqbracketbothcanvaliddefinitqbracketsgqbinomialusesexpressionsrepresentlastqbinomialgngqfactorialqfactorialqfactorialgqgammatwzgexamplreadqsummplppackaghypergeometricsummatcocopyrightharaldboeingwolframkoepuniversitkasselgrocedurqsimpcombprovidsimplificatmechanismallthosunctionagfqgfalsoqhypertermqpsihyperterm[{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Help H eading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 " " 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0ents which both represent any valid algebraic expression. It is defined as" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "qGAMMA(z,q)=qpochhamme r(q,q,infinity)/qpochhammer(q^z,q,infinity)*(1-q)^(1-z)" "6#/-%'qGAMMA G6$%\"zG%\"qG*(-%,qpochhammerG6%F(F(%)infinityG\"\"\"-F+6%)F(F'F(F-!\" \"),&F.F.F(F2,&F.F.F'F2F." }{TEXT -1 1 "." }}}}{SECT 0 {PARA 0 "" 0 " " {TEXT 26 9 "Examples:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boeing~&~Wolfram~Koep f,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The p rocedure qsimpcomb provides a simplification mechanism for all those f unctions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "qsimpcomb(qpochhammer( a,q,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&\"\"\"!\"\"%\"aGF%F%,& F%F&*&F'F%%\"qGF%F%F%,&F%F&*&F'F%)F*\"\"#F%F%F%,&F%F&*&F'F%)F*\"\"$F%F %F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPsnomial function uses three arguments, where the first two are any valid algebraic expressi ons representing integer values (symbolic) and the last one is a name \+ or a name raised to an integer power. The function is defined as:" }} {PARA 259 "" 0 "" {XPPEDIT 18 0 "qbinomial(n,k,q) = qpochhammer(q,q,n) /(qpochhammer(q,q,k)*qpochhammer(q,q,n-k)) " "6#/-%*qbinomialG6%%\"nG% \"kG%\"qG*&-%,qpochhammerG6%F)F)F'\"\"\"*&-F,6%F)F)F(F.-F,6%F)F),&F'F. F(!\"\"F.F5" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 11 "qfactorial:" } {TEXT -1 222 " The qfactorial function has two arguments. The first on e is any valid algebraic expression that represents an integer value ( symbolic) and the second one is a name or a name raised to any integer power. It is defined as: " }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "qfactor ial(k,q) = qpochhammer(q,q,k)/(1-q)^k" "6#/-%+qfactorialG6$%\"kG%\"qG* &-%,qpochhammerG6%F(F(F'\"\"\"),&F-F-F(!\"\"F'F0" }}}{SECT 0 {PARA 0 " " 0 "" {TEXT 26 7 "qGAMMA:" }{TEXT -1 108 " The qGAMMA function has tw o argumnteger value (symbolic). The function is defined as: " }} {PARA 257 "" 0 "" {XPPEDIT 18 0 "qpochhammer(a,q,infinity)=product(1-a *q^j,j=0..infinity)" "6#/-%,qpochhammerG6%%\"aG%\"qG%)infinityG-%(prod uctG6$,&\"\"\"F.*&F'F.)F(%\"jGF.!\"\"/F1;\"\"!F)" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 2 "6#/-%,qpochhamm erG6%%\"aG%\"qG%\"kG-%*PIECEWISEG6%7$-%(productG6$,&\"\"\"F2*&F'F2)F(% \"jGF2!\"\"/F5;\"\"!,&F)F2F6F22F9F)7$F2/F)F97$-F/6$*$F1F6/F5;F)F22F)F9 " }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 5 "qfac:" }{TEXT -1 81 " This f unction is a synonym for qpochhammer, i.e. qfac(a,q,k)=qpochhammer(a,q ,k)." }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "qbrackets:" }{TEXT -1 105 " The qbrackets function has two arguments. Both can be any valid \+ algebraic expression. The definition is:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "qbrackets(k,q) = (q^k-1)/(q-1)" "6#/-%*qbracketsG6$%\"k G%\"qG*&,&)F(F'\"\"\"F,!\"\"F,,&F(F,F,F-F-" }}}{SECT 0 {PARA 0 "" 0 " " {TEXT 26 10 "qbinomial:" }{TEXT -1 233 " The qbi59 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Help for:" }{TEXT -1 7 " B asic " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 10 "-functions" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling Sequences:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 110 "qpochhammer(a,q,infinity);\nqpochhammer(a,q, k);\nqfac(a,q,k);\nqbrackets(k,q);\nqfactorial(k,q);\nqbinomial(n,k,q) ;" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 13 "Description: " }}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "qpochhammer:" }{TEXT 2 284 " The qpochham mer function takes three arguments. The first parameter is any algebra ic expression, the second one is of type name or name raised ot an int eger power and the third variable is an algebraic expression represent ing an iJibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpeadingnormalheadlistitembulletfunctqsimpcombsimplificatqghypergeometrictermqratioratiocallsequencnqratioparameteralgebraicexpressnknamedescripttriesimplifinvolvpowerqbracketqbinomialqfactorialqgammaqpochhammqfunctreturnequalfgkgwrtintegvariablrgsuchwhererationaldesigndecidwhethrationaltypeworkfinehowevarbitrarmightbutmayfailrationalformcaseapplicatqsimplifyqsimplifproducexpectresultqratioshortcutsubsfunctionalitbuiltinexpandextendbasicfunctionbinomialthususealsoyieldsimplifiexamplreadqsummplppackagsummatcocopyrightharaldboeingwolframkoepfuniversitkasselgnqqpochhammergagngbinomialgqpochhammeragfkgfnoteqpochhammernqpochhammngfqhypertermqpsihyperterm[{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Help H eading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 " " 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0tio(f,k)" }{TEXT -1 19 " is a sho rtcut for " }{TEXT 19 26 "qsimpcomb(subs(k=k+1,f)/f)" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 86 "The functionality of the builtin Maple function expand is extended to simplify \"basic " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 16 "-functions\" (as " }{HYPERLNK 17 "qbrackets, q binomial, qfactorial, qGAMMA qpochhammer" 2 "qfunctions" "" }{TEXT -1 19 "). Thus the use of " }{TEXT 19 9 "expand(f)" }{TEXT -1 4 " or " } {TEXT 19 18 "normal(f,expanded)" }{TEXT -1 30 " also yields simplified forms." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boe ing~&~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "f:= 1/qpochhammer(a,q,-n)*(-q/a)^n*\nq^binomial(n,2 ) /qpochhammer(q/a,q,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGqinteger variable) if there is a function " }{XPPEDIT 18 0 "r(k)" "6 #-%\"rG6#%\"kG" }{TEXT -1 10 " such that" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "f(k+1) = r(k) * f(k)" "6#/-%\"fG6#,&%\"kG\"\"\"F)F)*&-% \"rG6#F(F)-F%6#F(F)" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 6 "wh ere " }{XPPEDIT 18 0 "r(k)" "6#-%\"rG6#%\"kG" }{TEXT -1 16 " is ration al in " }{XPPEDIT 18 0 "q^k" "6#)%\"qG%\"kG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 152 "qsimpcomb was designed to decide whether the \+ ratio f(k+i)/f(k) for an integer i is rational. For this type of terms it works fine. However an arbitrary " }{XPPEDIT 18 0 "q" "6#%\"qG" } {TEXT -1 26 "-hypergeometric term wrt. " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 24 " might be rational wrt. " }{XPPEDIT 18 0 "q^k" "6#)%\"qG %\"kG" }{TEXT -1 95 " but qsimpcomb may fail to simplify it to this ra tional form. In such a case an application of " }{HYPERLNK 17 "qsimpli fy" 2 "qsimplify" "" }{TEXT -1 35 " might produce the expected result. " }}{PARA 15 "" 0 "" {TEXT 19 11 "qraerms\n qra tio - Simplification of " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 27 "- hypergeometric term ratios" }}{PARA 4 "" 0 "" {TEXT 26 18 "Calling Seq uences:" }}{PARA 256 "" 0 "" {TEXT -1 26 "qsimpcomb(f);\nqratio(f,k); " }}{PARA 4 "" 0 "" {TEXT 26 11 "Parameters:" }}{PARA 257 "" 0 "" {TEXT -1 40 "f - an algebraic expression\nk - a name" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 13 "The function " }{TEXT 18 9 "qsimpcomb" }{TEXT -1 19 " tries to \+ simplify " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 47 "-hypergeometric \+ expressions, involving powers, " }{HYPERLNK 17 "qbrackets, qbinomial's , qfactorial's, qGAMMA's and qpochhammer's" 2 "qfunctions" "" }{TEXT -1 46 " and returns an expression that is equal to f." }}{PARA 15 "" 0 "" {TEXT -1 9 "The term " }{XPPEDIT 18 0 "f(k)" "6#-%\"fG6#%\"kG" } {TEXT -1 11 " is called " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 24 "- hypergeometric wrt. to " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 46 " ( an  0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 14 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT 257 2 " \+ " }{TEXT -1 30 "qsimpcomb - Simplification of " }{XPPEDIT 18 0 "q" "6 #%\"qG" }{TEXT -1 71 "-hypergeometric tJibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpeadingnormalheadlistitembulletfunctqsimpcombsimplificatqghypergeometrictermqratioratiocallsequencnqratioparameteralgebraicexpressnknamedescripttriesimplifinvolvpowerqbracketqbinomialqfactorialqgammaqpochhammqfunctreturnequalfgkgwrtintegvariablrgsuchwhererationaldesigndecidwhethrationaltypeworkfinehowevarbitrarmightbutmayfailrationalformcaseapplicatqsimplifyqsimplifproducexpectresultqratioshortcutsubsfunctionalitbuiltinexpandextendbasicfunctionbinomialthususealsoyieldsimplifiexamplreadqsummplppackagsummatcocopyrightharaldboeingwolframkoepfuniversitkasselgnqqpochhammergagngbinomialgqpochhammeragfkgfnoteqpochhammernqpochhammngfqhypertermqpsihypertermp!{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Help H eading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 " " 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 **-% ,qpochhammerG6%%\"aG%\"qG,$%\"nG!\"\"F-),$*&F*\"\"\"F)F-F-F,F1)F*-%)bi nomialG6$F,\"\"#F1-F'6%F0F*F,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "qsimpcomb(subs(n=n+1,f)/f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "qsimpcomb(qpochh ammer(a*q,q,k+1)/qpochhammer(a,q,k-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,&\"\"\"!\"\"*&%\"aGF%)%\"qG%\"kGF%F%F%,&F%F&*(F(F%F)F%F*F%F%F %,&F*F&F'F%F%,&F%F&F(F%F&F*F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 " Note: f is equal to one, but qsimpcomb fails to simplify this term." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f:= 1/qpochhammer(a,q,2*n)*qpochha mmer(a,q^2,n)*\nqpochhammer(a*q,q^2,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(-%,qpochhammerG6%%\"aG%\"qG,$*&\"\"#\"\"\"%\"nGF.F.!\"\" -F'6%F)*$)F*F-F.F/F.-F'6%*&F)F.F*F.F3F/F." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "qsimpcomb(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*( -%,qpochhammerG6%%\"aG%\"qG,$*&\"\"#\"\"\"%\"nGF,F,!\"\"-F%6%F'*$)F(F+ F,F-F,-F%6%*&F'F,F(F,F1Fotio(f,k)" }{TEXT -1 19 " is a sho rtcut for " }{TEXT 19 26 "qsimpcomb(subs(k=k+1,f)/f)" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 86 "The functionality of the builtin Maple function expand is extended to simplify \"basic " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 16 "-functions\" (as " }{HYPERLNK 17 "qbrackets, q binomial, qfactorial, qGAMMA qpochhammer" 2 "qfunctions" "" }{TEXT -1 19 "). Thus the use of " }{TEXT 19 9 "expand(f)" }{TEXT -1 4 " or " } {TEXT 19 18 "normal(f,expanded)" }{TEXT -1 30 " also yields simplified forms." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boe ing~&~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "f:= 1/qpochhammer(a,q,-n)*(-q/a)^n*\nq^binomial(n,2 ) /qpochhammer(q/a,q,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGinteger variable) if there is a function " }{XPPEDIT 18 0 "r(k)" "6 #-%\"rG6#%\"kG" }{TEXT -1 10 " such that" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "f(k+1) = r(k) * f(k)" "6#/-%\"fG6#,&%\"kG\"\"\"F)F)*&-% \"rG6#F(F)-F%6#F(F)" }{TEXT -1 1 "," }}{PARA 14 "" 0 "" {TEXT -1 6 "wh ere " }{XPPEDIT 18 0 "r(k)" "6#-%\"rG6#%\"kG" }{TEXT -1 16 " is ration al in " }{XPPEDIT 18 0 "q^k" "6#)%\"qG%\"kG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 152 "qsimpcomb was designed to decide whether the \+ ratio f(k+i)/f(k) for an integer i is rational. For this type of terms it works fine. However an arbitrary " }{XPPEDIT 18 0 "q" "6#%\"qG" } {TEXT -1 26 "-hypergeometric term wrt. " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 24 " might be rational wrt. " }{XPPEDIT 18 0 "q^k" "6#)%\"qG %\"kG" }{TEXT -1 95 " but qsimpcomb may fail to simplify it to this ra tional form. In such a case an application of " }{HYPERLNK 17 "qsimpli fy" 2 "qsimplify" "" }{TEXT -1 35 " might produce the expected result. " }}{PARA 15 "" 0 "" {TEXT 19 11 "qraerms\n qra tio - Simplification of " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 27 "- hypergeometric term ratios" }}{PARA 4 "" 0 "" {TEXT 26 18 "Calling Seq uences:" }}{PARA 256 "" 0 "" {TEXT -1 26 "qsimpcomb(f);\nqratio(f,k); " }}{PARA 4 "" 0 "" {TEXT 26 11 "Parameters:" }}{PARA 257 "" 0 "" {TEXT -1 40 "f - an algebraic expression\nk - a name" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 13 "The function " }{TEXT 18 9 "qsimpcomb" }{TEXT -1 19 " tries to \+ simplify " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 47 "-hypergeometric \+ expressions, involving powers, " }{HYPERLNK 17 "qbrackets, qbinomial's , qfactorial's, qGAMMA's and qpochhammer's" 2 "qfunctions" "" }{TEXT -1 46 " and returns an expression that is equal to f." }}{PARA 15 "" 0 "" {TEXT -1 9 "The term " }{XPPEDIT 18 0 "f(k)" "6#-%\"fG6#%\"kG" } {TEXT -1 11 " is called " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 24 "- hypergeometric wrt. to " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 46 " ( an proceedprocesprodproduproducproductW productg3profprogram proof' proposprov7provid?pspsipsigptionpuiseux pute puting qbinomialC) qbinomialgqbracketC* qbracketsg#qcharli  qcharlierg qfacqfactor qfactorial? qfactorialgqfunct/qgg             qgamma3%qgf/qgos qgosp, {PARA 260 "" 0 "" {XPPEDIT 18 0 "qpochhammer(a,q,n+k) = qpochhammer(a,q,n) * qpochhamm er(a*q^n,q,k)" "6#/-%,qpochhammerG6%%\"aG%\"qG,&%\"nG\"\"\"%\"kGF+*&-F %6%F'F(F*F+-F%6%*&F'F+)F(F*F+F(F,F+" }{TEXT -1 1 " " }}{PARA 15 "" 0 " " {TEXT 26 11 "Rule (iii):" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "qpochha mmer(a,q,-n) = 1 / qpochhammer(a*q^(-n),q,n)" "6#/-%,qpochhammerG6%% \"aG%\"qG,$%\"nG!\"\"*&\"\"\"F--F%6%*&F'F-)F(,$F*F+F-F(F*F+" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 26 10 "Rule (iv):" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "qpochhammer(a*q^(-k*n),q,n) = (-a)^n * q^(binomial (n,2)-k*n^2) * qpochhammer(q/a,q,k*n) / qpochhammer(q/a,q,k*n-n)" "6#/ -%,qpochhammerG6%*&%\"aG\"\"\")%\"qG,$*&%\"kGF)%\"nGF)!\"\"F)F+F/**),$ F(F0F/F))F+,&-%)binomialG6$F/\"\"#F)*&F.F)*$F/F9F)F0F)-F%6%*&F+F)F(F0F +*&F.F)F/F)F)-F%6%*&F+F)F(F0F+,&*&F.F)F/F)F)F/F0F0" }{TEXT -1 1 " " }} {PARA 15 "" 0 "" {TEXT 26 9 "Rule (v):" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "qpochhammer(a*q^(k*n),q,n) = qpochhammer(a,q,(k+1)*n) / qpochh ammer(a,q,{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 38 "It might help to use assumptions (see " }{HYPERLNK 17 "assume" 2 "assume" "" } {TEXT -1 108 ") on variables. qsimplify can only take advantage of the type integer or any subtype, e.g. posint or negint." }}{PARA 15 "" 0 "" {TEXT 262 12 "Restrictions" }{TEXT -1 26 " for the following rules: " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k " "6#%\"kG" }{TEXT -1 14 " are integers." }}{PARA 15 "" 0 "" {TEXT 26 9 "Rule (o):" }{TEXT -1 11 " The basic " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 117 "-functions such as qbinomial, qbrackets and qfactorial \+ are converted into terms involving only qpochhammer functions." }} {PARA 15 "" 0 "" {TEXT 26 9 "Rule (i):" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "qpochhammer(a,1/q,n) = qpochhammer(1/a,q,n) * (-a)^n / q^binom ial(n,2)" "6#/-%,qpochhammerG6%%\"aG*&\"\"\"F)%\"qG!\"\"%\"nG*(-F%6%*& F)F)F'F+F*F,F)),$F'F+F,F))F*-%)binomialG6$F,\"\"#F+" }{TEXT -1 1 " " } }{PARA 15 "" 0 "" {TEXT 26 10 "Rule (ii):" }}9 "-hypergeometr ic expressions\n combine - Combining qpochhammer fu nctions" }}{PARA 4 "" 0 "" {TEXT 26 18 "Calling Sequences:" }}{PARA 256 "" 0 "" {TEXT -1 37 "qsimplify(f);\ncombine(f,qpochhammer);" }} {PARA 4 "" 0 "" {TEXT 26 11 "Parameters:" }}{PARA 257 "" 0 "" {TEXT -1 27 "f - an algebraic expression" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 56 "qsimplify is desi gned as a simplification procedure for " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 34 "-hypergeometric expressions where " }{HYPERLNK 17 "qsimp comb" 2 "qsimpcomb" "" }{TEXT -1 111 " delivers unsatisfying results. \+ It applies Rule (o) to Rule (viii) stated below recursively to f. If y ou assign" }}{PARA 258 "" 0 "" {TEXT 19 20 "infolevel[qsum]:= 5;" }} {PARA 14 "" 0 "" {TEXT -1 61 "qsimplify will show most of the substitu tions that were made." }}{PARA 15 "" 0 "" {TEXT -1 59 "If you want to \+ use only Rule (vii) and (viii), you can use " }{TEXT 19 22 "combine(f, qpochhammer)" }0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 14 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 14 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 14 266 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Function:" }{TEXT 257 23 " qsimplify - Simplify " }{TEXT 258 1 "q" }{TEXT 259 80 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } 3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 14 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 gibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpeadingnormaltextheadwarnerrorlistembulletitemfunctqgospanalogugospalgorithmcallsequencesnqgospnqgosperupperlowerqqparameteralgebraicexpressnqnamenlsummatrangnupplowergeneralhypergeometricnqqintegernzequationsoptionalargumentdescriptionimplementationqgdecidwhethtermfgkgantidifferencggexistqsimpcombfunctioupwardownwardhypergeomtricrespectiveyoucalreturnantidifferencdefaultwillupwardmaychangspecifyingargumdownatendparametalsoqsumsettequalsumotherwismessaglastshortcutentersummandbasicunctanyintegequivalqphihypertermprefactorcanomitalwaytypelefthandsideoptionrightcorrespondvalufollowingpossiblinformathowstandardwishapplsimplificatprocedursimpfunctakeresultachievsingsimpliffactorpowercombinavoidsimplifictusefalsnullernomighttryseritruethusallowsearchthusefulhypergeometriccontainadditionposposintpositGpositi possibilitpossiblSposterio posteriori powerc>  ppackagOr" }{TEXT -1 59 " returns no antidifference you might try to use the \+ option " }{TEXT 19 11 "series=true" }{TEXT -1 16 ", thus allowing " } {TEXT 19 7 "qgosper" }{TEXT -1 43 " to search for an antidifference th at is a " }{TEXT 269 1 "q" }{TEXT -1 45 "-hypergeometric series. This \+ is useful for a " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 72 "-hypergeo metric expression containing an additional parameter such that " } {TEXT 19 7 "qgosper" }{TEXT -1 9 " finds a " }{XPPEDIT 18 0 "q" "6#%\" qG" }{TEXT -1 90 "-hypergeometric antidifference whenever we substitut e the parameter by a positive integer." }}{PARA 15 "" 0 "" {TEXT -1 112 "The most time consuming part in qgosper and qsumrecursion consist s of solving a system of linear equations (see " }{HYPERLNK 17 "qsum[t heory]" 2 "qsum[theory]" "" }{TEXT -1 14 "). The option " }{TEXT 19 11 "solvemethod" }{TEXT -1 80 " determines which procedures are used f or solving this system and can be set to " }{TEXT 19 4 "auto" }{TEXT -1 2 ", " }{TEXT 19 3 "ABP" }{TEXr " }{TEXT 19 1 "f" }{TEXT -1 33 " is equal to one you can omit it." }}{PARA 15 "" 0 "" {TEXT 264 18 "Optional arguments" }{TEXT -1 20 " are always of type " }{TEXT 19 9 "name=name" }{TEXT -1 142 ", where the left-hand side is the name of the option and the right-hand side the corresponding value. The follo wing options are possible (see " }{HYPERLNK 17 "qsum[setting]" 2 "qsum [setting]" "" }{TEXT -1 54 " for informations how to change the standa rd setting)." }}{PARA 15 "" 0 "" {TEXT -1 50 "If you wish to apply the simplification procedure " }{TEXT 19 8 "simpfunc" }{TEXT -1 91 " that takes one argument to the result of qgosper you could achieve it by u sing the option " }{TEXT 19 17 "simplify=simpfunc" }{TEXT -1 128 ". By default the result will be factorized and power terms will be combine d if possible. To avoid any simplifiction you can use " }{TEXT 19 14 " simplify=false" }{TEXT -1 5 " (or " }{TEXT 19 13 "simplify=NULL" } {TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXT -1 3 "If " }{TEXT 19 7 "qgosp elt it will return an upward antidifference. This may be changed by speci fying the optional argument " }{TEXT 19 19 "antidifference=down" } {TEXT -1 44 " at the end of the parameter list (see also " }{HYPERLNK 17 "qsum[setting]" 2 "qsum[setting]" "" }{TEXT -1 2 ")." }}{PARA 15 " " 0 "" {TEXT -1 24 "If qgosper is called by " }{TEXT 19 19 "qgosper(f, q,k=l..h)" }{TEXT -1 40 " it will return a term that is equal to " } {TEXT 19 13 "sum(f,k=l..h)" }{TEXT -1 7 ", if a " }{TEXT 268 1 "q" } {TEXT -1 35 "-hypergeometric antidifference for " }{TEXT 19 1 "f" } {TEXT -1 48 " exists. Otherwise an error message is returned." }} {PARA 15 "" 0 "" {TEXT -1 139 "The last two calling sequences are shor tcuts for entering the summand of a generalized basic hypergeometric f unction, i.e. for any integer " }{TEXT 19 1 "i" }{TEXT -1 10 " the cal l " }{TEXT 19 30 "qgosper(f,upper,lower,q^i,z,k)" }{TEXT -1 18 " is eq uivalent to " }{TEXT 19 49 "qgosper(f*qphihyperterm(upper,lower,q^i,z, k),q,k)" }{TEXT -1 19 ". If the prefactoetric term " }{XPPEDIT 18 0 "f(k)" "6#- %\"fG6#%\"kG" }{TEXT -1 35 " a q-hypergeometric antidifference " } {XPPEDIT 18 0 "g(k)" "6#-%\"gG6#%\"kG" }{TEXT -1 13 " exists (see " } {HYPERLNK 17 "qsimpcomb" 2 "qsimpcomb" "" }{TEXT -1 16 "). The functio n " }{XPPEDIT 18 0 "g(k)" "6#-%\"gG6#%\"kG" }{TEXT -1 26 " is an upwar d or downward " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 35 "-hypergeome tric antidifference for " }{XPPEDIT 18 0 "f(k)" "6#-%\"fG6#%\"kG" } {TEXT -1 3 " if" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "f(k) = g(k+1) - g( k)" "6#/-%\"fG6#%\"kG,&-%\"gG6#,&F'\"\"\"F-F-F--F*6#F'!\"\"" }{TEXT -1 14 " or " }{XPPEDIT 18 0 "f(k) = g(k) - g(k-1)" "6#/-%\"f G6#%\"kG,&-%\"gG6#F'\"\"\"-F*6#,&F'F,F,!\"\"F0" }}{PARA 14 "" 0 "" {TEXT -1 13 "respectively." }}{PARA 15 "" 0 "" {TEXT -1 12 "If you cal l " }{TEXT 19 14 "qgosper(f,q,k)" }{TEXT -1 24 " then qgosper returns \+ a " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 35 "-hypergeometric antidif ference for " }{TEXT 19 1 "f" }{TEXT -1 123 " if it exists. By defauosper(f,q,k=a..b,...); \n\nqgosper(f,[q1,q2,...],k,...);\nqgo sper(f,[q1,q2,...],k=l..h,...);\n\nqgosper(f,upper,lower,qq,z,k,...); \+ \nqgosper(f,upper,lower,qq,z,k=l..h,...);\nqgosper(upper,lower,qq,z,k, ...); " }}{PARA 4 "" 0 "" {TEXT 26 11 "Parameters:" }}{PARA 257 "" 0 " " {TEXT -1 236 "f - an algebraic expression\nq, q1, q2 - \+ names\nl, h - algebraic expressions, the summation range\nupper, lo wer - lists of algebraic expressions, the upper and the lower\n \+ parameters of the general " }{TEXT 267 1 "q" } {TEXT -1 145 "-hypergeometric function\nqq - a name or a name^inte ger\nz - an algebraic expression\n... - sequence of equatio ns, optional arguments" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 12 "Descri ption:" }}{PARA 15 "" 0 "" {TEXT -1 42 "This function is an implementa tion of the " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 59 "-analogue of \+ Gosper's algorithm that decides whether for a " }{XPPEDIT 18 0 "q" "6# %\"qG" }{TEXT -1 21 "-hypergeomsunsparcsolarimaplinputcourimathtimehyperlinkoutputnormalheadbulletitempreolhelveticafontformalpowerserusagcallsequencformalpowerserexpreqndirvarordermethodfpsparametexpressequatnameoptionalirectleftrightrealcomplexarbutrecommendsummatvariablresultupperbounddesearchedhypergeometricrationalexplikinfodescriptfunctcanexpandmeromorphicfunctioncertaintypeintotheircorrespondlaurpuiseuxseriumtermforminfinitsymmetrnumbshiftointdevelopmfollowsupportnoeithhaveerivatsomewherintegfunctionsatisflinearhomogeneoudifferialwithconstantcoefficientformalpowersertriefindformalpowererieexpansrespectatpointalsoworkcaselogarithmicessentialsingularitfirstlookomogeneoudifferentialpolynomialhencderivatmustknownbelowyouexamplunidentificomputgivenargumcontroldepthsearchhighvaluwillincreaschancsolutcomplexitwelldefaultcomputeasymptoticmayaroundinfinityexperfesultinfinitysqsimp qsimpcombKN  qsimpli qsimplifqsuqsum[q qsumdiff qsumdiffeq qsumrqsumrecu  qsumrecur  qsumrecurs/1 qsumrecursiqualquat rarahmanrailraisramrameterran rang#ratrath", " } {HYPERLNK 17 "qsum" 2 "qsum" "" }{TEXT 261 2 ", " }{HYPERLNK 17 "qsum[ theory]" 2 "qsum[certificate]" "" }{TEXT 265 2 ", " }{HYPERLNK 17 "qre csolve" 2 "qrecsolve" "" }{TEXT 262 2 ", " }{HYPERLNK 17 "sum2qhyper" 2 "sum2qhyper" "" }{TEXT 263 9 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 } }{TEXT -1 48 "-Charlier polynomials, Koekoek, Swarttouw (3.23)" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 61 "qsumrecursion([q^(-n),q^(-x)],[0],q,-q^(n+1)/a ,qCharlier(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*(%\"aG\"\"\")% \"qG,&%\"xGF'F'F'F'-%*qCharlierG6#%\"nGF'!\"\"*&,,)F),(F+F'F'F'F/F'F'* &F&F')F),&F+F'\"\"#F'F'F')F),$*&F8F'F/F'F'F0*&F&F'F3F'F0*&F&F'F(F'F'F' -F-6#,&F/F'F'F0F'F'**,&)F)F/F'*&F&F'F)F'F'F')F)F+F',&FCF0F)F'F'-F-6#,& F/F'F8F0F'F0\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" }{TEXT 256 1 " " }{HYPERLNK 17 "qfunctions" 2 "qfunctions" "" }{TEXT 266 2 ", " }{HYPERLNK 17 "qgosper" 2 "qgosper" "" }{TEXT 260 2 -n)], [q^(alpha+1)], q, -x*q^(n+al pha+1), qLaguerre(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*(%\"qG\" \"\",&)F&%\"nGF'F'!\"\"F'-%*qLaguerreG6#F*F'F+*&,,*$)F&\"\"#F'F+*&)F&, &*&F3F'F*F'F'%&alphaGF'F'%\"xGF'F'F&F+)F&,(F*F'F8F'F'F'F')F&,&F*F'F'F' F'F'-F-6#,&F*F'F'F+F'F'*(,&F&F')F&,&F8F'F*F'F+F'F&F'-F-6#,&F*F'F3F+F'F '\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 48 "-Charlier polynomials, Koekoek, Swarttouw (3.23)" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 61 "qsumrecursion([q^(-n),q^(-x)],[0],q,-q^(n+1)/a ,qCharlier(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*(%\"aG\"\"\")% \"qG,&%\"xGF'F'F'F'-%*qCharlierG6#%\"nGF'!\"\"*&,,)F),(F+F'F'F'F/F'F'* &F&F')F),&F+F'\"\"#F'F'F')F),$*&F8F'F/F'F'F0*&F&F'F3F'F0*&F&F'F(F'F'F' -F-6#,&F/F'F'F0F'F'**,&)F)F/F'*&F&F'F)F'F'F')F)F+F',&FCF0F)F'F'-F-6#,& F/F'F8F0F'F0\"\"!" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" }{TEXT 256 1 " " }{HYPERLNK 17 "qfunctions" 2 "qfunctions" "" }{TEXT 266 2 ", " }{HYPERLNK 17 "qgosper" 2 "qgosper" "" }{TEXT 260 2 orde r. In this case qsumrecursion returns a list containing the " } {XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 58 "-hypergeometric term as a fi rst entry and restrictions on " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 21 " as the second entry." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "E xamples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%PPackage~\"q-Hypergeometric~Summ ation\",~Maple~V-8G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%coCopyright~19 98-2002,~Harald~Boeing~&~Wolfram~Koepf,~University~of~KasselG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "qsumrecursion(qpochhammer(q^ (-n),q,k)*z^k/qpochhammer(q,q,k),q,k,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&)%\"qG%\"nG\"\"\"-%\"SG6#F(F)F)*&,&F&!\"\"%\"zGF)F)-F+6#,&F (F)F)F/F)F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "q" "6#%\"qG " }{TEXT -1 48 "-Laguerre polynomials, Koekoek, Swarttouw (3.21)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "qsumrecursion(qpochhammer(q^(alpha +1),q,n)/qpochhammer(q,q,n),\n [q^(l necessary informations to prove the resul ting recurrence equation and stores those in the global variable " } {TEXT 19 20 "_qsumrecursion_proof" }{TEXT -1 31 ". For further informa tions see " }{HYPERLNK 17 "qsum[theory]" 2 "qsum[theory]" "" }{TEXT -1 54 ". By default no proof informations will be calculated." }} {PARA 15 "" 0 "" {TEXT -1 11 "The option " }{TEXT 19 5 "proof" }{TEXT -1 135 " can be set to any (unassigned) name.\nIf specified, a certifi cate will be calculated and the information which is stored by default in " }{TEXT 19 20 "_qsumrecursion_proof" }{TEXT -1 129 " will be copi ed to that name. The default direction of the certificate is upwards, \+ unless the option certificate was set to down." }}{PARA 15 "" 0 "" {TEXT -1 11 "The option " }{TEXT 19 15 "rec2qhyper=true" }{TEXT -1 25 " can be used to return a " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 195 "-hypergeometric term solution for the sum (instead of a recurrenc e equation) if qsumrecursion found a recurrence equation of first  of qsumrecu rsion contain explicit summation bounds which are not equal to " } {TEXT 19 19 "-infinity..infinity" }{TEXT -1 31 ". (To change this beha vior see " }{HYPERLNK 17 "qsum[setting]" 2 "qsum[setting]" "" }{TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXT -1 11 "The option " }{TEXT 19 11 "i nhomo2homo" }{TEXT -1 15 " can be set to " }{TEXT 19 4 "true" }{TEXT -1 4 " or " }{TEXT 19 5 "false" }{TEXT -1 13 ", default is " }{TEXT 19 5 "false" }{TEXT -1 16 ".\nIf the option " }{TEXT 19 3 "rhs" } {TEXT -1 11 " is set to " }{TEXT 19 4 "true" }{TEXT -1 63 ", the resul ting recurrence equation might be inhomogeneous. If " }{TEXT 19 11 "in homo2homo" }{TEXT -1 11 " is set to " }{TEXT 19 4 "true" }{TEXT -1 84 ", qsumrecursion will try to convert this recurrence equation into a h omogeneous one." }}{PARA 15 "" 0 "" {TEXT -1 11 "The option " }{TEXT 19 11 "certificate" }{TEXT -1 15 " can be set to " }{TEXT 19 4 "down" }{TEXT -1 4 " or " }{TEXT 19 2 "up" }{TEXT -1 153 ".\nIf specified, qs umrecursion will provide alp leads to another linear system of equati ons. This seems to be slower for most examples.)" }}{PARA 15 "" 0 "" {TEXT -1 11 "The option " }{TEXT 19 8 "recorder" }{TEXT -1 57 " may be set to a posint or a range of posint, default is " }{TEXT 19 4 "1..5 " }{TEXT -1 143 ".\nIf a range is specified qsumrecursion will start t o look for a recurrence equation in that range. A positive integer n i s equivalent to n..n." }}{PARA 15 "" 0 "" {TEXT -1 11 "The option " } {TEXT 19 3 "rhs" }{TEXT -1 15 " can be set to " }{TEXT 19 4 "true" } {TEXT -1 4 " or " }{TEXT 19 5 "false" }{TEXT -1 282 ".\nIt determines \+ whether for the conversion of the recurrence equation for the input te rm into one for the corresponding sum is done by calculating the right -hand side, i.e. the inhomogeneous part of the recurrence equation, or just by assuming finite support of the input term (see " }{HYPERLNK 17 "qsum[theory]" 2 "qsum[theory]" "" }{TEXT -1 150 "). By default the right-hand side will only be calculated if the parametersld choo se the fastest method." }}{PARA 15 "" 0 "" {TEXT -1 11 "The option " } {TEXT 19 16 "gausselim_normal" }{TEXT -1 15 " can be set to " }{TEXT 19 6 "factor" }{TEXT -1 2 ", " }{TEXT 19 6 "normal" }{TEXT -1 2 ", " } {TEXT 19 14 "normalexpanded" }{TEXT -1 43 " and other procedures that \+ are returned by " }{TEXT 19 29 "`gausseliminate/procedures()`" }{TEXT -1 13 "; default is " }{TEXT 19 6 "normal" }{TEXT -1 13 ".\nWhen using " }{TEXT 19 21 "solvemethod=gausselim" }{TEXT -1 2 ", " }{TEXT 19 16 "gausselim_normal" }{TEXT -1 98 " determines which procedure is used f or simplification during the process of Gaussian elimination." }} {PARA 15 "" 0 "" {TEXT -1 11 "The option " }{TEXT 19 9 "recursion" } {TEXT -1 15 " may be set to " }{TEXT 19 4 "down" }{TEXT -1 4 " or " } {TEXT 19 2 "up" }{TEXT -1 13 ", default is " }{TEXT 19 4 "down" } {TEXT -1 243 ".\nIt controls whether the recurrence equation returned \+ by qsumrecursion is downwards or upwards. (We recommend to use the def ault value as switching to u,~Universit y~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "re:= q^n*f( n+3)-q^3*q^(2*n)*f(n+2)-\n(q^(2*n)+q)*f(n+1)+q*q^n*(q^(2*n)+q)*f(n)=0; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#reG/,**&)%\"qG%\"nG\"\"\"-%\"fG 6#,&F*F+\"\"$F+F+F+*()F)F0F+)F),$*&\"\"#F+F*F+F+F+-F-6#,&F*F+F6F+F+!\" \"*&,&F3F+F)F+F+-F-6#,&F*F+F+F+F+F:**F)F+F(F+F " 0 "" {MPLTEXT 1 0 44 "qrecsolve(re,q,f(n),output=q hypergeometric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$)%\"qG,&%\"nG \"\"\"-%)binomialG6$F(\"\"#F)1\"\"!F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "re:= f(n+2)-(1+q)*f(n+1)+q*(1-q*q^(2*n))*f(n)=0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#reG/,(-%\"fG6#,&%\"nG\"\"\"\"\"#F,F ,*&,&F,F,%\"qGF,F,-F(6#,&F+F,F,F,F,!\"\"*(F0F,,&F,F,*&F0F,)F0,$*&F-F,F +F,F,F,F4F,-F(6#F+F,F,\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "qrecsolve(re,q,f(n),split=false);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "qrecsolve(re,q,f( n));" }}{PARA 12 ""g]" "" }{TEXT -1 1 ")" }}{PARA 15 "" 0 "" {TEXT -1 11 "The option " }{TEXT 19 5 "split" }{TEXT -1 252 " con trols to some extent the introduction of additional algebraic numbers \+ which are required to solve the recurrence equation. The leading and t railing coefficients of the recursion have to be factored completely i n order to obtain all solutions. With " }{TEXT 19 5 "split" }{TEXT -1 8 " set to " }{TEXT 19 5 "false" }{TEXT -1 57 " only rational factoriz ation is used. Thus by specifying " }{TEXT 19 11 "split=false" }{TEXT -1 144 " the procedure might run faster but some solutions - which req uire the introduction of algebraic numbers - may be lost. By default i t is set to " }{TEXT 19 4 "true" }{TEXT -1 1 "." }}}{SECT 0 {PARA 4 " " 0 "" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%PPackage~\"q -Hypergeometric~Summation\",~Maple~V-8G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boeing~&~Wolfram~KoepfqG" }{TEXT -1 26 "-hypergeometric terms. If " }{TEXT 19 9 "downra tio" }{TEXT -1 4 " or " }{TEXT 19 7 "upratio" }{TEXT -1 23 " is specif ied, not the " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 22 "-hypergeomet ric terms " }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#%\"nG" }{TEXT -1 38 " are returned but rather their ratios " }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#% \"nG" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "f(n-1)" "6#-%\"fG6#,&%\"nG\"\"\" F(!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "f(n+1)" "6#-%\"fG6#,&%\"nG \"\"\"F(F(" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#%\"nG" } {TEXT -1 13 ".\nWhen using " }{TEXT 19 22 "output=qhypergeometric" } {TEXT -1 99 " qrecsolve returns a list of lists where each sublist con tains two entries: The first element is a " }{XPPEDIT 18 0 "q" "6#%\"q G" }{TEXT -1 100 "-hypergeometric solution of the recurrence equation \+ and the second element contains restrictions on " }{XPPEDIT 18 0 "n" " 6#%\"nG" }{TEXT -1 32 ".\n(For a global change see also " }{HYPERLNK 17 "qsum[setting]" 2 "qsum[settin" {TEXT -1 6 "where " } {XPPEDIT 18 0 "a[k]" "6#&%\"aG6#%\"kG" }{TEXT -1 6 " is a " }{TEXT 263 1 "q" }{TEXT -1 28 "-hypergeometric term w.r.t. " }{XPPEDIT 18 0 " k" "6#%\"kG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 11 "The optio n " }{TEXT -1 81 "output determines in which way qrecsolve returns the solutions and may be set to " }{TEXT 19 9 "downratio" }{TEXT -1 2 ", \+ " }{TEXT 19 7 "upratio" }{TEXT -1 2 ", " }{TEXT 19 7 "downrec" }{TEXT -1 2 ", " }{TEXT 19 5 "uprec" }{TEXT -1 4 " or " }{TEXT 19 15 "qhyperg eometric" }{TEXT -1 26 ". By default it is set to " }{TEXT 19 5 "uprec " }{TEXT -1 109 " meaning that a homogeneous upward recurrence equatio n of first order is returned for each possible solution " }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#%\"nG" }{TEXT -1 79 " (i.e. a homogeneous recur rence equation is returned). If the option is set to " }{TEXT 19 15 "q hypergeometric" }{TEXT -1 15 " all solutions " }{XPPEDIT 18 0 "f(n)" " 6#-%\"fG6#%\"nG" }{TEXT -1 17 " are returned as " }{XPPEDIT 18 0 "q" " 6#%\" n exists, an empty list is returned. Otherwise a list is returned wher e each solution is stored in another list." }}{PARA 15 "" 0 "" {TEXT -1 60 "The recurrence equation may be homogeneous or inhomogeneous." } }{PARA 15 "" 0 "" {TEXT -1 11 "The option " }{TEXT 19 8 "solution" } {TEXT -1 40 " determines for which sort of solutions " }{TEXT 18 9 "qr ecsolve" }{TEXT -1 25 " looks. It can be set to " }{TEXT 19 10 "polyno mial" }{TEXT -1 2 ", " }{TEXT 19 8 "rational" }{TEXT -1 2 ", " }{TEXT 19 15 "qhypergeometric" }{TEXT -1 4 " or " }{TEXT 19 6 "series" } {TEXT -1 85 ". By default it is set to qhypergeometric, i.e. the proce dure tries to determine all " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 39 "-hypergeometric solutions. When set to " }{TEXT 19 6 "series" } {TEXT -1 44 " qrecsolve returns all solutions of the form" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "f(n)=sum(a[k]*(q^n)^k,k=0..infinity)" "6#/- %\"fG6#%\"nG-%$sumG6$*&&%\"aG6#%\"kG\"\"\"))%\"qGF'F/F0/F/;\"\"!%)infi nityG" }{TEXT -1 1 "," }}{PARA 14 "" 0 " 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Function:" }{TEXT 257 23 " qrecsolve - Find all " }{TEXT 261 1 "q" }{TEXT 262 50 "-hypergeometr ic solutions of a recurrence equation" }}{PARA 4 "" 0 "" {TEXT 26 16 " Calling Sequence" }{TEXT 260 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 25 "qr ecsolve(re,q,f(n),...);" }}{PARA 4 "" 0 "" {TEXT 26 11 "Parameters:" } }{PARA 257 "" 0 "" {TEXT -1 203 "re - an algebraic expression or an e quation\nq - a name\nf - a name, the recurrence function\nn - a \+ name, the recurrence variable\n... - a sequence of equations, optional arguments (of type name=name)" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 91 "This procedure decid es whether a given recurrence equation with polynomial coefficients in " }{XPPEDIT 18 0 "q^n" "6#)%\"qG%\"nG" }{TEXT -1 9 " has any " } {XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 26 "-hypergeometric solutions " }{XPPEDIT 18 0 "f(n)" "6#-%\"fG6#%\"nG" }{TEXT -1 127 ". If no solutiotibmintelntmaplinputcourimathtimehyperlinkcommoutputhelpheadnormalbulletitemdashfunctsumqhypwrithypergeometricgeneralbasichypergeometriccallsequencsuparameteralgebraicexpressnqnamenknamesummatvariabldescriptprocuregiveninfinitqhypertermwrtreturnrrormessagotherwiswithentrphiphiguglgqgzgfirstlistrepresupperlowerionthirdjustnamlastrepresentevaluatpointexamplreadqsummplppackagcocopyrightharaldboeingwolframkoepfuniversitkasselgqbinomialngqsimpcombqpochhammergqgfkgqpochhammcgagnqpochhammqpochhammqpochhammererbgngfalsoqsumrecurs#{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Help H eading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 " " 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0ultumumg ummatumrecurs ununappunassignunct' unctionunderundeterm undetermin unevaluatuni#  unidentifi uniquunivuniversi  universitIunles unsatisfyup7upper+upratio uprec upwarupward3ure uresurlurrencursion ususausag&roced ure writes a given sum sum(" }{TEXT 264 1 "f" }{TEXT 267 1 "," } {TEXT 269 2 " k" }{TEXT 270 15 " = 0..infinity)" }{TEXT 268 1 " " } {TEXT -1 51 " as generalized basic hypergeometric function (see " } {HYPERLNK 17 "qhyperterm" 2 "qhyperterm" "" }{TEXT -1 6 "). If " } {TEXT 265 1 "f" }{TEXT -1 8 " is not " }{TEXT 263 1 "q" }{TEXT -1 21 " -hypergeometric wrt. " }{TEXT 266 1 "k" }{TEXT -1 78 " it returns an e rror message. Otherwise it returns a function with 4 entries: " } {XPPEDIT 18 0 "phi(u,l,q,z)" "6#-%$phiG6&%\"uG%\"lG%\"qG%\"zG" }} {PARA 258 "" 0 "" {TEXT -1 112 "The first two are lists and represent \+ the upper and lower list of the generalized basic hypergeometric funct ion," }}{PARA 259 "" 0 "" {TEXT -1 34 "the third entry is just the nam e q" }}{PARA 260 "" 0 "" {TEXT -1 79 "and the last one is an algebraic expression, representing the evaluation point." }}}{SECT 0 {PARA 4 " " 0 "" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {X -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }0 0 0 -1 -1 -1 3 24 0 0 0 0 -1 0 }{PSTYLE "" 16 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 3 36 0 0 0 0 -1 0 }{PSTYLE "" 16 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 3 36 0 0 0 0 -1 0 } {PSTYLE "" 16 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 3 36 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Function:" }{TEXT 257 23 " sum2qhyper - Write a " }{TEXT 261 1 "q" }{TEXT 262 64 "-hypergeometr ic sum as generalized basic hypergeometric function" }}{PARA 4 "" 0 " " {TEXT 26 18 "Calling Sequences:" }}{PARA 256 "" 0 "" {TEXT -1 20 "su m2qhyper(f, q, k);" }}{PARA 4 "" 0 "" {TEXT 26 11 "Parameters:" }} {PARA 257 "" 0 "" {TEXT -1 76 "f - an algebraic expression\nq - a na me\nk - a name, the summation variable" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 39 "This p>5{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Help H eading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 " " 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{C intdiffeqrodriguesrecursionrodriguesdiffeq GFrecursion GFdiffeqqqsum qsum,theoryqsum,setting qfunctions qpochhammerqfac qbrackets qfactorialqGAMMA qbinomial qsimpcombqratio qsimplify qgosper qhypertermqphihypertermqpsihypertermqsumrecursion qrecsolve sum2qhyper qsumdiffeqFPSFormalPowerSeries FPS,FPSFPS,FormalPowerSeriesFPS,HolonomicDE SimpleDE StandardSumFPS,standardsum hsum,theory HolonomicDEFPSBriefdescriptionFPS,SimpleDE SimpleREFPS,SimpleRE standardsumhsumhsum,references simpcomb hyperterm kfreerec fasenmyergosperextended_gospersumrecursion closedform Closedform zeilbergerWZcertificate rec2poly recpoly rec2hyper rechyper Sumtohypersumdeltanablasumdelta+nabla sumdiffeq sumdiffrule sumintrule contgosperintrecursionPERLNK 17 "qsumrecursion" 2 "qsumrecursion" "" }{TEXT 266 6 " \+ " }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } PARA 11 "" 1 "" {XPPMATH 20 "6#%PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boeing ~&~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "qphihyperterm([a,b],[c],q,z,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,-%,qpochhammerG6%%\"aG%\"qG%\"kG\"\"\"-F%6%%\"bGF(F)F *)%\"zGF)F*-F%6%%\"cGF(F)!\"\"-F%6%F(F(F)F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "qpsihyperterm([a,b],[c],q,z,k);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*0-%,qpochhammerG6%%\"aG%\"qG%\"kG\"\"\"-F%6%%\"bGF(F )F*)!\"\",$F)F/F*)F(,$*(\"\"#F/F)F*,&F)F*F*F/F*F/F*)%\"zGF)F*-F%6%%\"c GF(F)F/-F%6%F(F(F)F/" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also :" }{TEXT 256 2 " " }{HYPERLNK 17 "qsimpcomb" 2 "qsimpcomb" "" } {TEXT 258 2 ", " }{HYPERLNK 17 "qsum" 2 "qsum" "" }{TEXT 259 2 ", " } {HY>5{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Help H eading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 " " 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{Clateral basic hypergeometric serie s." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `qsum6.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%PPackage~\"q-Hypergeometric~Summation\",~Maple~V-8G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%coCopyright~1998-2002,~Harald~Boeing ~&~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "qphihyperterm([a,b],[c],q,z,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,-%,qpochhammerG6%%\"aG%\"qG%\"kG\"\"\"-F%6%%\"bGF(F)F *)%\"zGF)F*-F%6%%\"cGF(F)!\"\"-F%6%F(F(F)F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "qpsihyperterm([a,b],[c],q,z,k);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*0-%,qpochhammerG6%%\"aG%\"qG%\"kG\"\"\"-F%6%%\"bGF(F )F*)!\"\",$F)F/F*)F(,$*(\"\"#F/F)F*,&F)F*F*F/F*F/F*)%\"zGF)F*-F%6%%\"c GF(F)F/-F%6%F(F(F)F/" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also :" }{TEXT 256 2 " " }{HYPERLNK 17 "qsimpcomb" 2 "qsimpcomb" "" } {TEXT 258 2 ", " }{HYPERLNK 17 "qsum" 2 "qsum" "" }{TEXT 259 2 ", " } {HYq,z, k)" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 264 39 "bilateral basic hypergeometric function" }{TEXT -1 1 " " } {XPPEDIT 18 0 "psi(U,L,q,z)" "6#-%$psiG6&%\"UG%\"LG%\"qG%\"zG" }{TEXT -1 16 " is defined as:\n" }{XPPEDIT 18 0 "psi(U,L,q,z)=sum(product(qpo chhammer(U[i],q,k),i=1..r)/product(qpochhammer(L[i],q,k),i=1..s)*z^k*( (-1)^k*q^binomial(k,2))^(s-r),k=-infinity..infinity)" "6#/-%$psiG6&%\" UG%\"LG%\"qG%\"zG-%$sumG6$**-%(productG6$-%,qpochhammerG6%&F'6#%\"iGF) %\"kG/F7;\"\"\"%\"rGF;-F06$-F36%&F(6#F7F)F8/F7;F;%\"sG!\"\")F*F8F;)*&) ,$F;FFF8F;)F)-%)binomialG6$F8\"\"#F;,&FEF;F " 0 "" {MPLTEXT 1 0 53 "qsumdiffeq([q^(-n),y],[0],q,-q^(n+1)/a,qCharlier(y)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*(%\"qG\"\"\",&)F&%\"nGF'F'!\" \"F'-%*qCharlierG6#%\"yGF'F'*(,&F'F+F&F'F',,*&F/F')F&\"\"#F'F+*&F/F'F& F'F+*(F&F'F)F'F/F'F'F&F'%\"aGF'F'-&%#DqGF.6#F,F'F'*,F/F')F1F5F',&F'F+F 6F'F'F&F'-&F;6$F/F/F " 0 "" {MPLTEXT 1 0 122 "qsumdiffeq(qpochhammer(q^(alpha+1),q,n)/qpochhammer(q,q,n),\n \+ [q^(-n)], [q^(alpha+1)], q, -x*q^(n+alpha+1), qLaguerre(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(**,&)%\"qG%\"nG\"\"\"F*!\"\"F*)F(%&alphaG F*F(F*-%*qLaguerreG6#%\"xGF*F**(,&F*F+F(F*F*,,**F1F*F'F*F,F*F(F*F*F*F* *&F,F*F(F*F+*(F1F*F,F*)F(\"\"#F*F+*(F1F*F,F*F(F*F+F*-&%#DqGF06#F.F*F** .,&F*F**&F1F*F(F*F*F*F1F*)F3F9F*F,F*F(F*-&F=6$F1F1F>F*F+\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 48 "-Char lier polynomials, Koekoek, Swarttouw (3.23)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "qsumdiffeq([q^(-n),y],[0],q,-q^(n+1)/a,qCharlier(y)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*(%\"qG\"\"\",&)F&%\"nGF'F'!\" \"F'-%*qCharlierG6#%\"yGF'F'*(,&F'F+F&F'F',,*&F/F')F&\"\"#F'F+*&F/F'F& F'F+*(F&F'F)F'F/F'F'F&F'%\"aGF'F'-&%#DqGF.6#F,F'F'*,F/F')F1F5F',&F'F+F 6F'F'F&F'-&F;6$F/F/F " 0 "" {MPLTEXT 1 0 35 "qsumdiffeq([ x,b,c],[d,e],q,k,S(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**,)%\"qG \"\"#\"\"\",&F)!\"\"%\"cGF)F),&F)F+%\"bGF)F)%\"kGF)-%\"SG6#%\"xGF)F)** ,&F)F+F'F)F),B*,F/F)F3F)F.F)F,F)F&F)F)**F/F)F.F)F,F)F'F)F+**F/F)F3F)F. F)F'F)F+**F/F)F3F)F,F)F'F)F+*,F/F)F3F)F.F)F,F)F'F)F)%\"dGF)F)F+%\"eGF) *&FF+F=F)FF+F)-&FH6%F3F3F3FIF)F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 48 "-Laguerre psedthusspecifymightrunfastrequirelosttrueexamplreadmplppackagsummatcocopyrightharaldboeingwolframkoepfuniversitkasselgregbinomialgsgiggfqpochhammergqsumrecursqsumrecursvreparameteralgebraicexpressquatnqnamenfnnvariablequationoptionalargumenttypedescriptprocedurdecideswhethgivenwithpolynomialcoefficientqgnganyhypergeometricfgnosolutioexistemptreturnotherwiswhereachsolutstoranothmayhomogeneouinhomogeneouoptiondeterminsortlookcansetpolynomialrationalqhypergeometricseridefaultprocduretrieformsuminfinitsumgagkgqgfinfinitygtermoptiowaydownratioupratiodownrecuprecqhypergeometricmeanupwardequatiofirstorderpossiblrecurrencdownratiospecifiedhypergeometricbutraththeirrationwhenusingsublistcontainentrelemsecondcontainrestrictglobalchangalsoqsumsettsplittrolsomeextentintroductadditionalnumberrequirsolvleadrailrecurshavefactorcompleteobtainfalsonlyfactorizationuum[theory]" 2 "qsum[theory]" "" }{TEXT -1 14 "). The o ption " }{TEXT 19 11 "solvemethod" }{TEXT -1 80 " determines which pro cedures are used for solving this system and can be set to " }{TEXT 19 4 "auto" }{TEXT -1 2 ", " }{TEXT 19 3 "ABP" }{TEXT -1 2 ", " } {TEXT 19 9 "gausselim" }{TEXT -1 4 " or " }{TEXT 19 5 "solve" }{TEXT -1 13 ", default is " }{TEXT 19 4 "auto" }{TEXT -1 78 ".\nIf solve is \+ specified, the builtin Maple procedure solve is used. If set to " } {TEXT 19 9 "gausselim" }{TEXT -1 91 ", Gaussian elimination is used (f irst proposed by P. Paule and A. Riese, see References in " } {HYPERLNK 17 "qsum" 2 "qsum" "" }{TEXT -1 9 "). Using " }{TEXT 19 3 "A BP" }{TEXT -1 81 ",