{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 14 0 0 0 0 0 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 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Cour ier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "E rror" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 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 "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 "" 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 "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 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 }} {SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "Tutorial ISSAC 2004, July 4, 2004" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 41 "Wolfram Koepf: Power Series and Summation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest art;" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "Computation of Power Se ries" }{TEXT 256 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Maple supp orts truncated power series" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "series(exp(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"\"\"!F%F%#F%\"\"#F(#F%\"\"'\"\"$#F%\"#C\" \"%#F%\"$?\"\"\"&-%\"OG6#F%F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "The following algorithm for the computation of Formal Power Series is from\nKoepf, Wolfram: Power Series in Computer Algebra, Journal of Sy mbolic Computation 13, 1992, 581-603" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "read \"FPS.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%OPackage~Formal~Power~Series,~Maple~V~-~Maple~8G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%TCopyright~1995,~Dominik~Gruntz,~Uni versity~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 14 "FPS(exp(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%*factorialG6#%\"kG!\"\")%\"xGF*\"\"\"/F*; \"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "infolev el[FPS]:=5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "FPS(exp(x),x );" }}{PARA 6 "" 1 "" {TEXT -1 39 "FPS/FPS: looking for DE of degree 1" }}{PARA 6 "" 1 "" {TEXT -1 36 "FPS/FPS: DE of degree 1 fou nd." }}{PARA 6 "" 1 "" {TEXT -1 16 "FPS/FPS: DE = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#F'G6#%\"xG\"\"\"-%\"FGF'!\"\"\"\"!" }}{PARA 6 "" 1 "" {TEXT -1 16 "FPS/FPS: RE = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#,&%\"kG\"\"\"F)F)*&-F%6#F(F)F'!\"\"" }}{PARA 6 "" 1 "" {TEXT -1 48 "FPS/hypergeomRE: RE is of hypergeometric type." }} {PARA 6 "" 1 "" {TEXT -1 44 "FPS/hypergeomRE: Symmetry number m := \+ 1" }}{PARA 6 "" 1 "" {TEXT -1 23 "FPS/hypergeomRE: RE: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"kG\"\"\"F'F'F'-%\"aG6#F%F'-F)6#F&" } }{PARA 6 "" 1 "" {TEXT -1 45 "FPS/hypergeomRE: RE valid for all k >= 0" }}{PARA 6 "" 1 "" {TEXT -1 29 "FPS/hypergeomRE: a(0) = 1" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%*factorialG6#%\"kG!\"\") %\"xGF*\"\"\"/F*;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "FPS(exp(x^2),x);" }}{PARA 6 "" 1 "" {TEXT -1 39 "FPS/ FPS: looking for DE of degree 1" }}{PARA 6 "" 1 "" {TEXT -1 36 "FP S/FPS: DE of degree 1 found." }}{PARA 6 "" 1 "" {TEXT -1 16 "FPS /FPS: DE = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#F'G6#%\"xG\"\" \"*(\"\"#F)F(F)-%\"FGF'F)!\"\"\"\"!" }}{PARA 6 "" 1 "" {TEXT -1 16 "FP S/FPS: RE = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#,&%\"kG\"\" \"F)F),$*(\"\"#F)-F%6#,&F(F)F)!\"\"F)F'F0F)" }}{PARA 6 "" 1 "" {TEXT -1 48 "FPS/hypergeomRE: RE is of hypergeometric type." }}{PARA 6 "" 1 "" {TEXT -1 44 "FPS/hypergeomRE: Symmetry number m := 2" }} {PARA 6 "" 1 "" {TEXT -1 23 "FPS/hypergeomRE: RE: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"kG\"\"\"\"\"#F'F'-%\"aG6#F%F',$*&F(F'-F*6#F& F'F'" }}{PARA 6 "" 1 "" {TEXT -1 46 "FPS/hypergeomRE: RE valid for a ll k >= -1" }}{PARA 6 "" 1 "" {TEXT -1 29 "FPS/hypergeomRE: a(0) \+ = 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%*factorialG6#%\" kG!\"\")%\"xG,$*&\"\"#\"\"\"F*F1F1F1/F*;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "a Puiseux series" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "FPS(exp(sqrt(x)),x);" }} {PARA 6 "" 1 "" {TEXT -1 39 "FPS/FPS: looking for DE of degree 1" }}{PARA 6 "" 1 "" {TEXT -1 39 "FPS/FPS: looking for DE of degree 2 " }}{PARA 6 "" 1 "" {TEXT -1 36 "FPS/FPS: DE of degree 2 found. " }}{PARA 6 "" 1 "" {TEXT -1 16 "FPS/FPS: DE = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(\"\"%\"\"\"%\"xGF'-%$F''G6#F(F'F'*&\"\"#F'-%#F'GF+ F'F'-%\"FGF+!\"\"\"\"!" }}{PARA 6 "" 1 "" {TEXT -1 16 "FPS/FPS: RE = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#,&%\"kG\"\"\"F)F),$*&#F) \"\"#F)*(-F%6#F(F)F'!\"\",&*&F-F)F(F)F)F)F)F1F)F)" }}{PARA 6 "" 1 "" {TEXT -1 48 "FPS/hypergeomRE: RE is of hypergeometric type." }} {PARA 6 "" 1 "" {TEXT -1 44 "FPS/hypergeomRE: Symmetry number m := \+ 1" }}{PARA 6 "" 1 "" {TEXT -1 23 "FPS/hypergeomRE: RE: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$**\"\"#\"\"\",&%\"kGF'F'F'F',&*&F&F'F)F'F 'F'F'F'-%\"aG6#F(F'F'-F-6#F)" }}{PARA 6 "" 1 "" {TEXT -1 46 "FPS/hyper geomRE: RE modified to k = 1/2*k" }}{PARA 6 "" 1 "" {TEXT -1 36 " FPS/hypergeomRE: => f := exp(x)" }}{PARA 6 "" 1 "" {TEXT -1 48 "F PS/hypergeomRE: RE is of hypergeometric type." }}{PARA 6 "" 1 "" {TEXT -1 44 "FPS/hypergeomRE: Symmetry number m := 2" }}{PARA 6 " " 1 "" {TEXT -1 23 "FPS/hypergeomRE: RE: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(,&%\"kG\"\"\"\"\"#F'F',&F&F'F'F'F'-%\"aG6#F%F'-F+6#F &" }}{PARA 6 "" 1 "" {TEXT -1 45 "FPS/hypergeomRE: RE valid for all \+ k >= 0" }}{PARA 6 "" 1 "" {TEXT -1 29 "FPS/hypergeomRE: a(0) = \+ 1" }}{PARA 6 "" 1 "" {TEXT -1 29 "FPS/hypergeomRE: a(1) = 1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$SumG6$*&-%*factorialG6#,$*&\"\"# \"\"\"%\"kGF.F.!\"\")%\"xGF/F./F/;\"\"!%)infinityGF.-F%6$*&-F)6#,&*&F- F.F/F.F.F.F.F0)F2,&F/F.#F.F-F.F.F3F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "FPS(arcsin(x),x);" }}{PARA 6 "" 1 "" {TEXT -1 39 "FPS /FPS: looking for DE of degree 1" }}{PARA 6 "" 1 "" {TEXT -1 39 "F PS/FPS: looking for DE of degree 2" }}{PARA 6 "" 1 "" {TEXT -1 36 "FPS/FPS: DE of degree 2 found." }}{PARA 6 "" 1 "" {TEXT -1 16 " FPS/FPS: DE = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&\"\"\"!\"\" *$)%\"xG\"\"#F'F'F'-%$F''G6#F+F'F'*&F+F'-%#F'GF/F'F'\"\"!" }}{PARA 6 " " 1 "" {TEXT -1 16 "FPS/FPS: RE = " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"aG6#,&%\"kG\"\"\"\"\"#F)**F(F*-F%6#F(F),&F(F)F)F)!\"\"F'F/" }} {PARA 6 "" 1 "" {TEXT -1 48 "FPS/hypergeomRE: RE is of hypergeometri c type." }}{PARA 6 "" 1 "" {TEXT -1 44 "FPS/hypergeomRE: Symmetry nu mber m := 2" }}{PARA 6 "" 1 "" {TEXT -1 23 "FPS/hypergeomRE: RE: \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*(,&%\"kG\"\"\"F(F(F(,&F'F(\"\" #F(F(-%\"aG6#F)F(!\"\",$*&)F'F*F(-F,6#F'F(F." }}{PARA 6 "" 1 "" {TEXT -1 45 "FPS/hypergeomRE: RE valid for all k >= 0" }}{PARA 6 "" 1 " " {TEXT -1 29 "FPS/hypergeomRE: a(0) = 0" }}{PARA 6 "" 1 "" {TEXT -1 44 "FPS/hypergeomRE: a(2*j) = 0 for all j>0." }}{PARA 6 "" 1 " " {TEXT -1 29 "FPS/hypergeomRE: a(1) = 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*,-%*factorialG6#,$*&\"\"#\"\"\"%\"kGF-F-F-)\" \"%,$F.!\"\"F--F(6#F.!\"#,&*&F,F-F.F-F-F-F-F2)%\"xGF6F-/F.;\"\"!%)infi nityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "infolevel[FPS]:=0: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "computation in steps" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f[0]:=a rcsin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"!-%'arcsinG6 #%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#),&%\"xG\"\"\"F&F&%\"kG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"!-%'arcsinG6#%\"xG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f[1]:=diff(f[0],x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"\"*&F'F'*$,&F'F'*$)%\"xG \"\"#F'!\"\"#F'F.F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "norm al(f[1]/f[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&,&F$F$*$ )%\"xG\"\"#F$!\"\"#F$F*-%'arcsinG6#F)F$F+" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "f[2]:=diff(f[1],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"#*&,&\"\"\"F**$)%\"xGF'F*!\"\"#!\"$F'F-F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ansatz:=sum(c[k]*f[k],k=0..2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ansatzG,(*&&%\"cG6#\"\"!\"\" \"-%'arcsinG6#%\"xGF+F+*&&F(6#F+F+,&F+F+*$)F/\"\"#F+!\"\"#F7F6F+*(&F(6 #F6F+F3#!\"$F6F/F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "nor mal(subs(c[0]=0,ansatz));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(&% \"cG6#\"\"\"!\"\"*&F&F))%\"xG\"\"#F)F)*&&F'6#F.F)F-F)F*F),&F)F)*$F,F)F *#!\"$F.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sol:=solve(no rmal(subs(c[0]=0,ansatz)),\{c[1],c[2]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<$/&%\"cG6#\"\"#F'/&F(6#\"\"\"*(F'F.%\"xGF.,&F.! \"\"*$)F0F*F.F.F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DE:=c[ 0]*F(x)+c[1]*diff(F(x),x)+c[2]*diff(F(x),x$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG,(*&&%\"cG6#\"\"!\"\"\"-%\"FG6#%\"xGF+F+*&&F(6#F+ F+-%%diffG6$F,F/F+F+*&&F(6#\"\"#F+-F46$F,-%\"$G6$F/F9F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "collect(numer(normal(subs(sol,c[0]= 0,DE/c[2]))),diff)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"xG\" \"\"-%%diffG6$-%\"FG6#F&F&F'F'*&,&F'!\"\"*$)F&\"\"#F'F'F'-F)6$F+-%\"$G 6$F&F3F'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "procedure comb ining these steps" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "DE:=HolonomicDE(arcsin(x),F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/,&*&%\"xG\"\"\"-%%diffG6$-%\"FG6#F(F(F)F)*(,&F(F )F)!\"\"F),&F(F)F)F)F)-F+6$F--%\"$G6$F(\"\"#F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(DE,F(x));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"FG6#%\"xG,&%$_C1G\"\"\"*&-%#lnG6#,&F'F**$,&F*!\" \"*$)F'\"\"#F*F*#F*F5F*F*%$_C2GF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "RE:=SimpleRE(arcsin(x),x,a(k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,&*(,&%\"kG\"\"\"F*F*F*,&F)F*\"\"#F*F*-%\"aG6#F+ F*!\"\"*&)F)F,F*-F.6#F)F*F*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rsolve(\{RE,a(0)=0,a(1)=1\},a(k));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%*PIECEWISEG6$7$\"\"!'%\"kG%%evenG7$**F)!\"\"%#PiG#F -\"\"#-%&GAMMAG6#,$*&F0F-F)\"\"\"F6F6-F26#,&*&F0F-F)F6F6#F6F0F6F-'F)%$ oddG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "some final examples: a La urent series" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "FPS(arcsin( x)^2/x^5,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*,-%*factoria lG6#%\"kG\"\"#)\"\"%F*\"\"\"-F(6#,&F.F.*&F+F.F*F.F.!\"\",&F*F.F.F.F3)% \"xG,&*&F+F.F*F.F.\"\"$F3F./F*;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "a complicated example that cannot be found in Gradsh teyn/Ryshik" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "FPS(exp(arcsin(x)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-% $SumG6$**,&*&\"\"%\"\"\")%\"kG\"\"#F+F+F+F+!\"\"-%(ProductG6$,&*&F*F+) %\"jGF.F+F+F+F+/F6;\"\"!F-F+-%*factorialG6#,$*&F.F+F-F+F+F/)%\"xGF=F+/ F-;F9%)infinityGF+-F%6$*,-F16$,(F+F+*&F.F+%\"jGF+F+*&F.F+)FKF.F+F+/FKF 8F+)F.F-F+,(*&F.F+F,F+F+*&F.F+F-F+F+F+F+F/-F;6#,&*&F.F+F-F+F+F+F+F/)F@ FUF+FAF+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "and an asymptotic ser ies" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "F PS((erf(x)-1)*exp(x^2),x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*&%#PiG#!\"\"\"\"#-%$SumG6$*,)F'%\"kG\"\"\"-%*factorialG6#,$*&F(F/F .F/F/F/)\"\"%,$F.F'F/-F16#F.F')*&F/F/%\"xGF',&*&F(F/F.F/F/F/F/F//F.;\" \"!%)infinityGF/F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Also covere d are holonomic special functions" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "FPS(LegendreP(n,x),x);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#,&*.\"\"#\"\"\"%#PiG#F&F%-%&GAMMAG6#,&F(F&*&F%!\"\"% \"nGF&F.F.-F*6#,$*&F%F.F/F&F&F.F/F.-%$SumG6$*,-%+pochhammerG6$,$*&F%F. F/F&F.%\"kGF&-F96$,&*&F%F.F/F&F&F(F&F=F&-%*factorialG6#,$*&F%F&F=F&F&F .)\"\"%F=F&)%\"xGFEF&/F=;\"\"!%)infinityGF&F&*,F%F&F'F(-F*6#F@F.-F*6#F ;F.-F56$*,-F96$F,F=F&-F96$,&F&F&*&F%F.F/F&F&F=F&FGF&-FC6#,&*&F%F&F=F&F &F&F&F.)FJFinF&FKF&F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "FP S(LegendreP(n,x),x=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*.) !\"\"%\"kG\"\"\")\"\"#,$F)F(F*-%+pochhammerG6$,&%\"nGF*F*F*F)F*-F/6$,$ F2F(F)F*-%*factorialG6#F)!\"#),&%\"xGF*F*F(F)F*/F);\"\"!%)infinityG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 47 "Computation of Holonomic Differential Equations" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Find a holonomic differential equa tion for f(x)=sin(x)*exp(x)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "f[0]:=sin(x)*exp(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"!*&-%$sinG6#%\"xG\"\"\"-%$expGF+F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f[1]:=diff(f[0],x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"\",&*&-%$cosG6#%\"xGF'-%$ expGF,F'F'*&-%$sinGF,F'F.F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "normal(f[1]/f[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%$cos G6#%\"xG\"\"\"-%$sinGF'F)F)F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f[2]:=diff(f[1],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%\"fG6#\"\"#,$*(F'\"\"\"-%$cosG6#%\"xGF*-%$expGF-F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ansatz:=expand(sum(c[k]*f[k],k=0..2 ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ansatzG,**(&%\"cG6#\"\"!\"\" \"-%$sinG6#%\"xGF+-%$expGF.F+F+*(&F(6#F+F+-%$cosGF.F+F0F+F+*(F3F+F,F+F 0F+F+**\"\"#F+&F(6#F9F+F5F+F0F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ansatz:=collect(ansatz,\{cos(x),sin(x)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ansatzG,&*&,&*&&%\"cG6#\"\"!\"\"\"-%$expG 6#%\"xGF-F-*&&F*6#F-F-F.F-F-F--%$sinGF0F-F-*&,&F2F-*(\"\"#F-&F*6#F:F-F .F-F-F--%$cosGF0F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "coe ffs(ansatz,\{cos(x),sin(x)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*& &%\"cG6#\"\"!\"\"\"-%$expG6#%\"xGF)F)*&&F&6#F)F)F*F)F),&F.F)*(\"\"#F)& F&6#F3F)F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sol:=solv e(\{coeffs(ansatz,\{cos(x),sin(x)\})\},\{c[0],c[1],c[2]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<%/&%\"cG6#\"\"!,$*&\"\"#\"\"\"&F(6#F -F.F./&F(6#F.,$*&F-F.F/F.!\"\"/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DE:=c[0]*F(x)+c[1]*diff(F(x),x)+c[2]*diff(F(x),x$2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG,(*&&%\"cG6#\"\"!\"\"\"-%\"F G6#%\"xGF+F+*&&F(6#F+F+-%%diffG6$F,F/F+F+*&&F(6#\"\"#F+-F46$F,-%\"$G6$ F/F9F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "DE:=collect(num er(normal(subs(sol,DE/c[0]))),diff)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/,(*&\"\"#\"\"\"-%\"FG6#%\"xGF)F)*&F(F)-%%diffG6$F*F-F)!\" \"-F06$F*-%\"$G6$F-F(F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:='f':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "Algebra of Holonomic Functions" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "read \"FPS.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%OPackage~Formal~Power~Series,~Maple~V~-~Map le~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%TCopyright~1995,~Dominik~Gru ntz,~University~of~BaselG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%^oCopyri ght~2002,~Detlef~M|gzller~&~Wolfram~Koepf,~University~of~KasselG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7P%(LaplaceG%.algebraicsubsG%.algeqtodiffeqG%.alge qtoseriesG%.algfuntoalgeqG%&borelG%.cauchyproductG%.diffeq*diffeqG%.di ffeq+diffeqG%,diffeqtableG%2diffeqtohomdiffeqG%,diffeqtorecG%)guesseqn G%(guessgfG%0hadamardproductG%0holexprtodiffeqG%)invborelG%,listtoalge qG%-listtodiffeqG%0listtohypergeomG%+listtolistG%.listtoratpolyG%*list torecG%-listtoseriesG%,maxdegcoeffG%*maxdegeqnG%,maxordereqnG%,mindegc oeffG%*mindegeqnG%,minordereqnG%*optionsgfG%,poltodiffeqG%)poltorecG%/ ratpolytocoeffG%(rec*recG%(rec+recG%,rectodiffeqG%,rectohomrecG%*recto procG%.seriestoalgeqG%/seriestodiffeqG%2seriestohypergeomG%-seriestoli stG%0seriestoratpolyG%,seriestorecG%/seriestoseriesG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The function sin(x)*exp(x), again:" }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The differential equa tion of sin(x):" }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "DE1:=diff(F(x),x$2)+F(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DE1G/,&-%%diffG6$-%\"FG6#%\"xG-%\"$G6$F-\"\"#\"\"\"F *F2\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The differential equa tion of exp(x):" }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "DE2:=diff(F(x),x)-F(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DE2G/,&-%%diffG6$-%\"FG6#%\"xGF-\"\"\"F*!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "`diffeq*diffeq`(DE1,DE2,F (x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"#\"\"\"-%\"FG6#%\"xGF &F&*&F%F&-%%diffG6$F'F*F&!\"\"-F-6$F'-%\"$G6$F*F%F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "and the sum sin(x)+exp(x) satisfies" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "`diffeq+diffeq`( DE1,DE2,F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%%diffG6$-%\"FG6# %\"xG-%\"$G6$F*\"\"$\"\"\"-F%6$F'F*F/-F%6$F'-F,6$F*\"\"#!\"\"F'F7" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Now a more complicated example: ex p(x)*Ai(x)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "DE1:=diff(F(x),x)-F(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ DE1G/,&-%%diffG6$-%\"FG6#%\"xGF-\"\"\"F*!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "DE2:=HolonomicDE(AiryAi(x),F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DE2G/,&-%%diffG6$-%\"FG6#%\"xG-%\"$G6$F- \"\"#\"\"\"*&F-F2F*F2!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "`diffeq*diffeq`(DE1,DE2,F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&\"\"\"F&%\"xG!\"\"F&-%\"FG6#F'F&F&-%%diffG6$F)-%\"$G6$F'\" \"#F&*&F2F&-F-6$F)F'F&F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "and t he sum exp(x)+Ai(x) satisfies" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "`diffeq+diffeq`(DE1,DE2,F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#<$/---%#@@G6$%\"DG\"\"#6#%\"FG6#\"\"!&%#_CGF.,**&,(%\"xG!\"\"\"\"\"F 7*$)F5F+F7F7F7-F-6#F5F7F7*&,&F5F7F8F6F7-%%diffG6$F:F5F7F7*&F5F7-F?6$F: -%\"$G6$F5F+F7F6*&,&F5F7F7F6F7-F?6$F:-FE6$F5\"\"$F7F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Similarly, HolonomicDE yields" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "HolonomicDE(exp( x)+AiryAi(x),F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&,(%\"xG!\" \"\"\"\"F)*$)F'\"\"#F)F)F)-%\"FG6#F'F)F)*(F'F),&F'F)F)F(F)-%%diffG6$F- F'F)F(*&F'F)-F36$F--%\"$G6$F'F,F)F(*&F1F)-F36$F--F96$F'\"\"$F)F)\"\"! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "Similar algorithms exist for sequences and recurrence equations. Assume we want to find a recurren ce equation w.r.t. k for" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "binomial(n,k)+binomial(k,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%) binomialG6$%\"nG%\"kG\"\"\"-F%6$F(F'F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The binomial coefficient S(k)=binomial(n,k) (first summ and) satisfies the equation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "S(k+1)/S(k)=expand(binomial(n,k+1)/binomial(n, k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"SG6#,&%\"kG\"\"\"F*F*F* -F&6#F)!\"\"*&,&%\"nGF*F)F-F*F(F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "w.r.t. k. This gives the holonomic recurrence equation " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "RE1:=collect(numer(normal(S( k+1)-expand(binomial(n,k+1)/binomial(n,k))*S(k))),S,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RE1G,&*&,&%\"kG\"\"\"F)F)F)-%\"SG6#F'F)F) *&,&F(F)%\"nG!\"\"F)-F+6#F(F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "The binomial coefficient S(k)=binomial(k,n) (second summand) sat isfies the equation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "S(k+1)/S(k)=expand(binomial(k+1,n)/binomial(k,n));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"SG6#,&%\"kG\"\"\"F*F*F*-F&6#F) !\"\"*&F(F*,(F)F*F*F*%\"nGF-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "w.r.t. k. This gives the holonomic recurrence equation " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "RE2:=collect(numer(normal(S(k+1)-ex pand(binomial(k+1,n)/binomial(k,n))*S(k))),S,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RE2G,&*&,(%\"kG\"\"\"F)F)%\"nG!\"\"F)-%\"SG6#,& F(F)F)F)F)F)*&,&F)F+F(F+F)-F-6#F(F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Therefore we get for the sum" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "`rec+rec`(RE1,RE2,S(k));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<%/-%\"SG6#\"\"\",&*&%\"nGF(&%#_CG6#\" \"!F(F(&F-F'F(/-F&F.,(F,F(*&F0F(F+F(!\"\"F0F(,(*&,<*&\"\"#F()%\"kG\"\" %F(F5*&\"\"*F()F<\"\"$F(F5*&\"#8F()F* &\"\"&F(FFF(F(*&F:F(F@F(F5*$)F+F=F(F(*&\"\")F(FDF(F5*&F:F(F+F(F5*&F=F( FHF(F5*(\"#7F(FDF(F+F(F5*(F\\oF(F*&F:F(F;F(F(*&\"#6F(F@F(F(*&\"#@F(FDF(F(*(F=F(F@F(F+F(F5*(FP F(FDF(F+F(F5*&FPF(FF(F " 0 "" {MPLTEXT 1 0 24 "`rec*rec`(RE1,RE2,S(k)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&%\"kG\"\"\"%\"nG!\"\"F'-%\" SG6#F&F'F'*&,(F&F'F'F'F(F)F'-F+6#,&F&F'F'F'F'F'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "Hype rgeometric Functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "hype rgeom([a,b],[c],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*hypergeomG6% 7$%\"aG%\"bG7#%\"cG%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sumtools[hyperterm]([a,b],[c],x,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*,-%+pochhammerG6$%\"aG%\"kG\"\"\"-F%6$%\"bGF(F))%\"xGF(F)-F%6$%\"cG F(!\"\"-%*factorialG6#F(F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "sum(sumtools[hyperterm]([a,b],[c],x,k),k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*hypergeomG6%7$%\"aG%\"bG7#%\"cG%\"xG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "hypergeom([a,b],[c],1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*hypergeomG6%7$%\"aG%\"bG7#%\"cG\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "simplify(hypergeom([a ,b],[c],1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**-%&GAMMAG6#%\"cG\"\" \"-F%6#,(F'F(%\"aG!\"\"%\"bGF-F(-F%6#,&F'F(F,F-F--F%6#,&F'F(F.F-F-" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 42 "Identification of Hypergeometric Functions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "We are interested in" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "s:=Sum((-1)^k/(2*k+1)!*x^ (2*k+1),k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG-%$Su mG6$*()!\"\"%\"kG\"\"\"-%*factorialG6#,&*&\"\"#F,F+F,F,F,F,F*)%\"xGF0F ,/F+;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F: =k->(-1)^k/(2*k+1)!*x^(2*k+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" FGf*6#%\"kG6\"6$%)operatorG%&arrowGF(*()!\"\"9$\"\"\"-%*factorialG6#,& *&\"\"#F0F/F0F0F0F0F.)%\"xGF4F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "r:=F(k+1)/F(k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"rG*.)!\"\",&%\"kG\"\"\"F*F*F*-%*factorialG6#,&*&\"\"#F*F)F*F*\"\"$F* F')%\"xGF.F*)F'F)F'-F,6#,&*&F0F*F)F*F*F*F*F*)F3F7F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&*&\"\"#\"\"\"%\"kGF(F(F'F(!\"\",&*&F'F(F)F(F(\"\"$F(F*%\"x GF'F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Hence" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "s=F(0)*hypergeom([],[3/ 2],-x^2/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*()!\"\"%\"kG \"\"\"-%*factorialG6#,&*&\"\"#F+F*F+F+F+F+F))%\"xGF/F+/F*;\"\"!%)infin ityG*&F3F+-%*hypergeomG6%7\"7##\"\"$F1,$*&\"\"%F)F3F1F)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Check" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "convert(s,hypergeom);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$sinG6#%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Maple simplifies completely, hence we don't see the hypergeometric form. The same applies to" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "simplify(x*hypergeom([],[3/2],-x^2/4));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "The following procedure uses the given algorithm and gives therefore the hypergeometric form:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sumtools[Sumtohyper](F(k),k) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\"-%*HypergeomG6%7\"7# #\"\"$\"\"#,$*&\"\"%!\"\"F$F-F1F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Another example" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "F:=binomial(n,k)*binomial(-n-1,k)*((1-x)/2)^k;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG*(-%)binomialG6$%\"nG%\"kG\"\"\" -F'6$,&F)!\"\"F+F/F*F+),&#F+\"\"#F+*&F3F/%\"xGF+F/F*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Sum(F,k=0..n)=sumtools[Sumtohyper]( F,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*(-%)binomialG6$%\" nG%\"kG\"\"\"-F)6$,&F+!\"\"F-F1F,F-),&#F-\"\"#F-*&F5F1%\"xGF-F1F,F-/F, ;\"\"!F+-%*HypergeomG6%7$,$F+F1,&F+F-F-F-7#F-F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Details of this algorithm and an implementation can \+ be found in the book" }}{PARA 0 "" 0 "" {TEXT -1 15 "Wolfram Koepf: " }{TEXT 260 24 "Hypergeometric Summation" }{TEXT -1 38 ", Vieweg, Braun schweig/Wiesbaden, 1998" }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "We can combine the FPS \+ and the identification algorithm:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s:=FPS(exp(x),x,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG-%$SumG6$*&-%*factorialG6#%\"kG!\"\")%\"xGF,\"\" \"/F,;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "op (1,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%*factorialG6#%\"kG!\"\") %\"xGF'\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "sumtools[S umtohyper](op(1,s),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*Hypergeom G6%7\"F&%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "s:='s':" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Write cos(x) in hypergeometric no tation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "fps:=FPS(cos(x), x,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fpsG-%$SumG6$*()!\"\"%\"kG \"\"\"-%*factorialG6#,$*&\"\"#F,F+F,F,F*)%\"xGF0F,/F+;\"\"!%)infinityG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sumtools[Sumtohyper](op (1,fps),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*HypergeomG6%7\"7##\" \"\"\"\"#,$*&\"\"%!\"\"%\"xGF*F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 64 "Computation of Recurrence Equations for Hypergeometric Functions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "How does one generate the result" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Sum(binomial(n,k), k=0..n)=\nsum(binomial(n,k),k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%$SumG6$-%)binomialG6$%\"nG%\"kG/F+;\"\"!F*)\"\"#F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "We do the following more complicated exam ple with Maple:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Sum(k*binomial(n,k),k=0..n)=\nsum(k*binomial(n,k),k=0 ..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&%\"kG\"\"\"-%)bin omialG6$%\"nGF(F)/F(;\"\"!F-,$*(\"\"#!\"\")F3F-F)F-F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "F:=(n,k)->k*binomial(n,k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6$%\"nG%\"kG6\"6$%)operatorG%& arrowGF)*&9%\"\"\"-%)binomialG6$9$F.F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "ansatz:=sum(sum(a(j,i)*F(n+j,k+i),i=0..1),j=0..1 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'ansatzG,**(-%\"aG6$\"\"!F*\" \"\"%\"kGF+-%)binomialG6$%\"nGF,F+F+*(-F(6$F*F+F+,&F,F+F+F+F+-F.6$F0F4 F+F+*(-F(6$F+F*F+F,F+-F.6$,&F0F+F+F+F,F+F+*(-F(6$F+F+F+F4F+-F.6$F " 0 "" {MPLTEXT 1 0 22 "ansatz:=ansatz/F(n,k) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'ansatzG*(,**(-%\"aG6$\"\"!F+\" \"\"%\"kGF,-%)binomialG6$%\"nGF-F,F,*(-F)6$F+F,F,,&F-F,F,F,F,-F/6$F1F5 F,F,*(-F)6$F,F+F,F-F,-F/6$,&F1F,F,F,F-F,F,*(-F)6$F,F,F,F5F,-F/6$F=F5F, F,F,F-!\"\"F.FC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ansatz:= expand(ansatz);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'ansatzG,8-%\"aG6 $\"\"!F)\"\"\"*(-F'6$F)F*F*,&%\"kGF*F*F*!\"\"%\"nGF*F**(F/F*F,F*F.F0F0 **F/F0F,F*F.F0F1F*F**&F,F*F.F0F0*(-F'6$F*F)F*,(F1F*F/F0F*F*F0F1F*F**&F 6F*F8F0F**(-F'6$F*F*F*F.F0F1F*F**&F;F*F.F0F***F/F0F;F*F.F0F1F*F**(F/F0 F;F*F.F0F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ansatz:=norma l(ansatz);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'ansatzG,$*(,@*&)%\"kG \"\"#\"\"\"-%\"aG6$\"\"!F0F,!\"\"*&F)F,-F.6$F0F,F,F,*(F-F,F*F,%\"nGF,F ,*(-F.6$F,F,F,F6F,F*F,F1**F+F,F3F,F6F,F*F,F1*&F8F,F*F,F1*(-F.6$F,F0F,F 6F,F*F,F,*&F=F,F*F,F,*&F-F,F*F,F,*&F*F,F3F,F1F8F,*&F3F,)F6F+F,F,*(F+F, F8F,F6F,F,*&F8F,FCF,F,*&F3F,F6F,F,F,,(F6F1F*F,F,F1F1F*F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ansatz:=numer(ansatz);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'ansatzG,@*&)%\"kG\"\"#\"\"\"-%\"aG6$\"\"!F.F *F**&F'F*-F,6$F.F*F*!\"\"*(F+F*F(F*%\"nGF*F2*(-F,6$F*F*F*F4F*F(F*F***F )F*F0F*F4F*F(F*F**&F6F*F(F*F**(-F,6$F*F.F*F4F*F(F*F2*&F;F*F(F*F2*&F+F* F(F*F2*&F(F*F0F*F*F6F2*&F0F*)F4F)F*F2*(F)F*F6F*F4F*F2*&F6F*FAF*F2*&F0F *F4F*F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqs:=\{coeffs(an satz,k)\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$eqsG<%,&-%\"aG6$\"\"! F*\"\"\"-F(6$F*F+!\"\",,*&-F(6$F+F+F+)%\"nG\"\"#F+F.*&F,F+F4F+F.F1F.*& F,F+F3F+F.*(F5F+F1F+F4F+F.,2F1F+*&F'F+F4F+F.*&F1F+F4F+F+*(F5F+F,F+F4F+ F+F,F+*&-F(6$F+F*F+F4F+F.F>F.F'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sol:=solve(eqs,\{seq(seq(a(j,i),j=0..1),i=0..1)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<&/-%\"aG6$\"\"\"\"\"!F+/-F(6 $F+F+,$*(,&%\"nGF*F*F*F*-F(6$F*F*F*F2!\"\"F5/-F(6$F+F*F//F3F3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "re:=sum(sum(a(j,i)*f(n+j,k+i ),i=0..1),j=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#reG,**&-%\"aG 6$\"\"!F*\"\"\"-%\"fG6$%\"nG%\"kGF+F+*&-F(6$F*F+F+-F-6$F/,&F0F+F+F+F+F +*&-F(6$F+F*F+-F-6$,&F/F+F+F+F0F+F+*&-F(6$F+F+F+-F-6$F " 0 "" {MPLTEXT 1 0 17 "re:=subs(sol,re);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#reG,(**,&%\"nG\"\"\"F)F)F)-%\"aG6$F)F)F)F (!\"\"-%\"fG6$F(%\"kGF)F-**F'F)F*F)F(F--F/6$F(,&F1F)F)F)F)F-*&F*F)-F/6 $F'F5F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "re:=numer(norm al(re/a(1,1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#reG,,*&-%\"fG6$% \"nG%\"kG\"\"\"F*F,!\"\"F'F-*&-F(6$F*,&F+F,F,F,F,F*F,F-F/F-*&-F(6$,&F* F,F,F,F1F,F*F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "RE:=sub s(\{seq(seq(f(n+j,k+i)=s(n+j),i=0..1),j=0..1)\},re);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG,(*(\"\"#\"\"\"-%\"sG6#%\"nGF(F,F(!\"\"*&F'F(F )F(F-*&-F*6#,&F,F(F(F(F(F,F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "RE:=map(factor,collect(RE,s))=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,&*(\"\"#\"\"\",&%\"nGF)F)F)F)-%\"sG6#F+F)!\"\"*&-F-6#F*F )F+F)F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Now we use the im plementation from the book" }}{PARA 0 "" 0 "" {TEXT -1 15 "Wolfram Koe pf: " }{TEXT 257 24 "Hypergeometric Summation" }{TEXT -1 38 ", Vieweg, Braunschweig/Wiesbaden, 1998" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "restart; read \"hsum9.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2004,~Wolfram~Koepf,~Univer sity~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "libname: =libname,\"C:/Dokumente und Einstellungen/koepf/Eigene Dateien/Koepf/M aple/Software/hsum\";" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(libnameG6$ Q9C:\\Programme\\Maple~9/lib6\"QhoC:/Dokumente~und~Einstellungen/koepf /Eigene~Dateien/Koepf/Maple/Software/hsumF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?hsum" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fasenmyer(k*binomial(n,k),k,s(n),1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"nG\"\"\"-%\"sG6#,&F&F'F'F'F'F'*(\"\"#F'-F)6#F&F'F+F'!\" \"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fasenmyer(binomi al(n,k)^2,k,s(n),1,1);" }}{PARA 8 "" 1 "" {TEXT -1 72 "Error, (in kfre erec) No kfree recurrence equation of order (1,1) exists\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fasenmyer(binomial(n,k)^2,k,s(n),2, 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&%\"nG\"\"\"\"\"#F(F(-%\" sG6#F&F(F(*(F)F(-F+6#,&F'F(F(F(F(,&*&F)F(F'F(F(\"\"$F(F(!\"\"\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fasenmyer(binomial(n-k,k),k ,s(n),2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%\"sG6#,&%\"nG\"\" \"\"\"#F*F*-F&6#F)!\"\"-F&6#,&F)F*F*F*F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "[seq(sum(binomial(n-k,k),k=0..n),n=0..10)]; n:=' n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"\"F$\"\"#\"\"$\"\"&\"\") \"#8\"#@\"#M\"#b\"#*)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "fa senmyer((-1)^k*binomial(n,k)^2,k,s(n),2,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&%\"nG\"\"\"\"\"#F(F(-%\"sG6#F&F(F(*(\"\"%F(-F+6# F'F(,&F'F(F(F(F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " fasenmyer(binomial(n,k)^3,k,s(n),2,1);" }}{PARA 8 "" 1 "" {TEXT -1 72 "Error, (in kfreerec) No kfree recurrence equation of order (2,2) exis ts\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fasenmyer(binomial( n,k)^3,k,s(n),3,1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**(,&*&\"\"$ \"\"\"%\"nGF)F)\"\"%F)F)),&F*F)F(F)\"\"#F)-%\"sG6#F-F)F)*(F.F),**&\"\" *F))F*F(F)F)*&\"#dF))F*F.F)F)*&\"$;\"F)F*F)F)\"#uF)F)-F06#,&F*F)F.F)F) !\"\"*(,&*&F(F)F*F)F)\"\"&F)F),(*&\"#:F)F9F)F)*&\"#bF)F*F)F)\"#[F)F)-F 06#,&F*F)F)F)F)F@**\"\")F),&*&F(F)F*F)F)\"\"(F)F))FMF.F)-F06#F*F)F@\" \"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Legendre polynomials" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Sum(bin omial(n,k)*binomial(-n-1,k)*((1-x)/2)^k,k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(-%)binomialG6$%\"nG%\"kG\"\"\"-F(6$,&F*!\"\" F,F0F+F,),&#F,\"\"#F,*&F4F0%\"xGF,F0F+F,/F+;\"\"!F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "This corresponds to the hypergeometric represen tation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Sumtohyper(binomial(n,k)*binomial(-n-1,k)*((1-x)/2)^k,k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%*HypergeomG6%7$,$%\"nG!\"\",&F(\"\" \"F+F+7#F+,&#F+\"\"#F+*&F/F)%\"xGF+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "fasenmyer(binomial(n,k)*binomial(-n-1,k)*((1-x)/2)^k, k,s(n),2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\" #F(F(-%\"sG6#F&F(F(*(%\"xGF(,&*&F)F(F'F(F(\"\"$F(F(-F+6#,&F'F(F(F(F(! \"\"*&F4F(-F+6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Com pute a three-term recurrence equation for the Laguerre polynomials." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "The Laguerre polynomials have th e representation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "LaguerreL(n,x)=\nsum((-1)^k/k!*binomial(n,k)*x^k,k=0. .infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*LaguerreLG6$%\"nG% \"xG-%*hypergeomG6%7#,$F'!\"\"7#\"\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Therefore we get" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "fasenmyer((-1)^k/k!*binomial(n,k)*x^k,k,s(n), 2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\"#F(F(-% \"sG6#F&F(F(*&,(*&F)F(F'F(F(%\"xG!\"\"\"\"$F(F(-F+6#,&F'F(F(F(F(F1*&F5 F(-F+6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "The general ized Laguerre polynomials have the hypergeometric representation" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Laguerr eL(n,alpha,x)=sum((-1)^k/k!*binomial(n+alpha,n-k)*x^k,k=0..infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*LaguerreLG6%%\"nG%&alphaG%\"xG *&-%)binomialG6$,&F'\"\"\"F(F/F'F/-%*hypergeomG6%7#,$F'!\"\"7#,&F(F/F/ F/F)F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Therefore we get" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "fasenmy er((-1)^k/k!*binomial(n+alpha,n-k)*x^k,k,s(n),2,1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\"#F(F(-%\"sG6#F&F(F(*&,*\"\"$F(% &alphaGF(*&F)F(F'F(F(%\"xG!\"\"F(-F+6#,&F'F(F(F(F(F3*&,(F'F(F0F(F(F(F( -F+6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Indefinite Summation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Indefinite sum of k*k!" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s:=sum(k*k!,k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG-%*factorialG6#%\"kG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Check:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "difference:=subs(k=k+1,s)-s;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%+differenceG,&-%*factorialG6#,&%\"kG \"\"\"F+F+F+-F'6#F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplify(difference);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"kG\"\" \"-%&GAMMAG6#,&F$F%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(difference-k*k!);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" !" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Maple's simplify treats bino mials etc. badly:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "simplify(binomial(n,k)/k!);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%&GAMMAG6#,&%\"nG\"\"\"F)F)F)-F%6#,&%\"kGF)F)F)!\"#- F%6#,(F(F)F)F)F-!\"\"F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We can check the algorithms internally used:" }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "infolevel[sum]:=5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sum((-1)^k*binomial(n,k),k);" }} {PARA 6 "" 1 "" {TEXT -1 36 "sum/indefnew: indefinite summation" }} {PARA 6 "" 1 "" {TEXT -1 81 "sum/extgosper: applying Gosper algorith m to a( k ):= (-1)^k*binomial(n,k)" }}{PARA 6 "" 1 "" {TEXT -1 65 "sum/gospernew/internal: a( k )/a( k -1):= (-n-1+k)/k " }}{PARA 6 "" 1 "" {TEXT -1 55 "sum/gospernew/internal: Gosper's al gorithm applicable" }}{PARA 6 "" 1 "" {TEXT -1 33 "sum/gospernew/inter nal: p:= 1" }}{PARA 6 "" 1 "" {TEXT -1 38 "sum/gospernew/internal: q:= -n-1+k" }}{PARA 6 "" 1 "" {TEXT -1 33 "sum/gospernew/internal : r:= k" }}{PARA 6 "" 1 "" {TEXT -1 43 "sum/gospernew/internal: \+ degreebound:= 0" }}{PARA 6 "" 1 "" {TEXT -1 53 "sum/gospernew/intern al: solving equations to find f" }}{PARA 6 "" 1 "" {TEXT -1 55 "sum/ gospernew/internal: Gosper's algorithm successful" }}{PARA 6 "" 1 " " {TEXT -1 36 "sum/gospernew/internal: f:= -1/n" }}{PARA 6 "" 1 " " {TEXT -1 45 "sum/indefnew: indefinite summation finished" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"kG\"\"\"%\"nG!\"\")F(F%F&-%)binomialG 6$F'F%F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(sumtools );" }}{PARA 7 "" 1 "" {TEXT -1 139 "Warning, these previously assigned names now have a global binding: Sumtohyper, extended_gosper, gosper, hyperterm, simpcomb, sumrecursion\n" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#7,%)HypersumG%+SumtohyperG%0extended_gosperG%'gosperG%/hyperrecursio nG%)hypersumG%*hypertermG%)simpcombG%-sumrecursionG%+sumtohyperG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gosper((-1)^k*binomial(n,k), k);" }}{PARA 6 "" 1 "" {TEXT -1 65 "sum/gospernew/internal: a( k \+ )/a( k -1):= (-n-1+k)/k" }}{PARA 6 "" 1 "" {TEXT -1 55 "sum/go spernew/internal: Gosper's algorithm applicable" }}{PARA 6 "" 1 "" {TEXT -1 33 "sum/gospernew/internal: p:= 1" }}{PARA 6 "" 1 "" {TEXT -1 38 "sum/gospernew/internal: q:= -n-1+k" }}{PARA 6 "" 1 " " {TEXT -1 33 "sum/gospernew/internal: r:= k" }}{PARA 6 "" 1 "" {TEXT -1 43 "sum/gospernew/internal: degreebound:= 0" }}{PARA 6 " " 1 "" {TEXT -1 53 "sum/gospernew/internal: solving equations to fin d f" }}{PARA 6 "" 1 "" {TEXT -1 55 "sum/gospernew/internal: Gosper's algorithm successful" }}{PARA 6 "" 1 "" {TEXT -1 36 "sum/gospernew/in ternal: f:= -1/n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"kG\"\" \"%\"nG!\"\")F(F%F&-%)binomialG6$F'F%F&F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 48 "Example from SIAM Reviews 36, 1994, Problem 94-2" } {MPLTEXT 1 0 0 "" }{TEXT 258 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Sum((-1)^(k+1)*(4*k+1)*(2*k)!/(k!*4^k*(2*k-1)*(k+1)!) ,k=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*0)!\"\", &%\"kG\"\"\"F+F+F+,&*&\"\"%F+F*F+F+F+F+F+-%*factorialG6#,$*&\"\"#F+F*F +F+F+-F06#F*F()F.F*F(,&*&F4F+F*F+F+F+F(F(-F06#F)F(/F*;F+%)infinityG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sum((-1)^(k+1)*(4*k+1)*(2* k)!/(k!*4^k*(2*k-1)*(k+1)!),k);" }}{PARA 6 "" 1 "" {TEXT -1 36 "sum/in defnew: indefinite summation" }}{PARA 6 "" 1 "" {TEXT -1 110 "sum/ex tgosper: applying Gosper algorithm to a( k ):= (-1)^(k+1)*(4*k +1)*(2*k)!/k!/(4^k)/(2*k-1)/(k+1)!" }}{PARA 6 "" 1 "" {TEXT -1 89 "sum /gospernew/internal: a( k )/a( k -1):= -1/2*(4*k+1)/(4*k- 3)/(k+1)*(2*k-3)" }}{PARA 6 "" 1 "" {TEXT -1 55 "sum/gospernew/interna l: Gosper's algorithm applicable" }}{PARA 6 "" 1 "" {TEXT -1 37 "sum /gospernew/internal: p:= 4*k+1" }}{PARA 6 "" 1 "" {TEXT -1 38 "sum /gospernew/internal: q:= -2*k+3" }}{PARA 6 "" 1 "" {TEXT -1 37 "su m/gospernew/internal: r:= 2*k+2" }}{PARA 6 "" 1 "" {TEXT -1 43 "su m/gospernew/internal: degreebound:= 0" }}{PARA 6 "" 1 "" {TEXT -1 53 "sum/gospernew/internal: solving equations to find f" }}{PARA 6 " " 1 "" {TEXT -1 55 "sum/gospernew/internal: Gosper's algorithm succe ssful" }}{PARA 6 "" 1 "" {TEXT -1 34 "sum/gospernew/internal: f:= \+ -1" }}{PARA 6 "" 1 "" {TEXT -1 45 "sum/indefnew: indefinite summatio n finished" }}{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 69 "sum((-1)^(k+1)*(4*k+1)*(2*k)!/(k!*4^k*(2*k-1)*(k+1)!),k=1..infinit y);" }}{PARA 6 "" 1 "" {TEXT -1 34 "sum/infinite: infinite summation " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "infolevel[sum]:=0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We do a more complicated example" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "s:=k!*binomial(n,k)/(n-k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG*(-%*factorialG6#%\"kG\"\"\"-% )binomialG6$%\"nGF)F*,&F.F*F)!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a:=subs(k=k+3,s)-s;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"aG,&*(-%*factorialG6#,&%\"kG\"\"\"\"\"$F,F,-%)binomialG6$%\"nGF* F,,(F1F,F+!\"\"F-F3F3F,*(-F(6#F+F,-F/6$F1F+F,,&F1F,F+F3F3F3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "b:=gosper(a,k);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"bG,$*.,(%\"nG!\"\"%\"kG\"\"\"\"\"$F+F+,:\" \"#F+*$)F(F,F+F)*(F,F+)F(F.F+F*F+F+*(F,F+F(F+)F*F.F+F)*(\"\"%F+F0F+F*F +F)*(\"\"'F+F2F+F4F+F+*(F6F+F(F+)F*F,F+F)*&F,F+F*F+F+*&F,F+F(F+F)*$F:F +F+*$)F(F6F+F+*$)F*F6F+F+F+,@F>F+*(F6F+F0F+F*F+F)*&F,F+F0F+F)*(F8F+F2F +F4F+F+*(\"\"*F+F2F+F*F+F+*&F.F+F2F+F+*(F6F+F(F+F:F+F)*(FGF+F(F+F4F+F) *(F6F+F(F+F*F+F)F@F+*&F,F+F:F+F+*&F.F+F4F+F+F(F)F*F+F,F+F),(F(F)F*F+F. F+F),(F(F)F*F+F+F+F),&*(-%*factorialG6#,&F*F+F,F+F+-%)binomialG6$F(FUF +,(F(F+F*F)F,F)F)F+*(-FS6#F*F+-FW6$F(F*F+,&F(F+F*F)F)F)F+F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "gosper(b,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "restart; read \"hsum9.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%V Package~\"Hypergeometric~Summation\",~Maple~V~-~Maple~9G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%YCopyright~1998-2004,~Wolfram~Koepf,~University ~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "s:=k!*binomi al(n,k)/(n-k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG*(-%*factorial G6#%\"kG\"\"\"-%)binomialG6$%\"nGF)F*,&F.F*F)!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a:=subs(k=k+3,s)-s;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,&*(-%*factorialG6#,&%\"kG\"\"\"\"\"$F,F,-%)binom ialG6$%\"nGF*F,,(F1F,F+!\"\"F-F3F3F,*(-F(6#F+F,-F/6$F1F+F,,&F1F,F+F3F3 F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "b:=gosper(a,k);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"bG,$*.,(%\"nG!\"\"%\"kG\"\"\"\"\"$ F+F+,:\"\"#F+*$)F(\"\"%F+F+*$)F(F,F+F)*$)F*F1F+F+*$)F*F,F+F+*(F1F+F3F+ 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 +*(F1F+F(F+F7F+F)*(F,F+F(F+F>F+F)F+,@F/F+*(F1F+F3F+F*F+F)*(FF +F+*(F1F+F(F+F7F+F)*&F,F+F3F+F)*(\"\"*F+F=F+F*F+F+*(FHF+F(F+F>F+F)F4F+ *&F,F+F7F+F+*&F.F+F=F+F+*(F1F+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),&*(-%*factorialG6#,&F*F+F,F+F+-%)binomial G6$F(FUF+,(F(F+F*F)F,F)F)F+*(-FS6#F*F+-FW6$F(F*F+,&F(F+F*F)F)F)F+F)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "gosper(b,k);" }}{PARA 8 " " 1 "" {TEXT -1 64 "Error, (in gosper) No hypergeometric term antidiff erence exists\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a:='a': \+ b:='b':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Gosper's Algorithm in Detail" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \"hsum9.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Maple~V~-~ Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2004,~Wol fram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "first example" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "a:=k*k!;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG*&% \"kG\"\"\"-%*factorialG6#F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rat:=subs(k=k+1,a)/a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ratG **,&%\"kG\"\"\"F(F(F(-%*factorialG6#F&F(F'!\"\"-F*6#F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rat:=normal(expand(rat));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ratG*&,(*$)%\"kG\"\"#\"\"\"F+*&F*F+F)F+F+ F+F+F+F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "q:=numer(r at);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG,(*$)%\"kG\"\"#\"\"\"F** &F)F*F(F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "r:=denom (rat);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG%\"kG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "q(k) and r( k+j) have a nontrivial gcd for j=1:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "gcd(q,subs(k=k+1,r));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&%\"kG\"\"\"F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "pqr:=update(p,subs(k=k-1,q),subs(k=k-1,r),k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pqrG7%%\"kGF&\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p:=op(1,pqr); q:=op(2,pqr); r:=op(3 ,pqr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG%\"kG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG%\"kG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"rG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:='f':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "RE:=subs(k=k+1,q)*f(k)-subs( k=k+1,r)*f(k-1)=p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,&*&,&%\" kG\"\"\"F*F*F*-%\"fG6#F)F*F*-F,6#,&F)F*F*!\"\"F1F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rsolve(RE,f(k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&,&F$!\"\"-%\"fG6#\"\"!F$F$-%&GAMMAG6#,&%\"k GF$\"\"#F$F'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=findf( p,q,r,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "s:=r/p*subs(k=k-1,f)*a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG-%*factorialG6#%\"kG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "second example" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a:=(-1)^k*binomial(n,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG*&)!\"\"%\"kG\"\"\"-%)binomialG6$%\"nG F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rat:=subs(k=k+1,a)/ a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ratG**)!\"\",&%\"kG\"\"\"F*F* F*-%)binomialG6$%\"nGF(F*)F'F)F'-F,6$F.F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rat:=normal(expand(rat));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ratG*&,&%\"nG!\"\"%\"kG\"\"\"F*,&F)F*F*F*F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "q:=numer(rat);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"qG,&%\"nG!\"\"%\"kG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "r:=denom(rat);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG,&%\"kG\"\"\"F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"\"\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "q(k) and r(k+j) have no nont rivial gcd for n a symbol, but for negative integer n. We will come ba ck to this case later." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "pqr:=update(p,subs(k=k-1,q),subs(k=k-1,r),k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pqrG7%\"\"\",(%\"nG!\"\"%\"kGF&F&F) F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p:=op(1,pqr); q:=op(2 ,pqr); r:=op(3,pqr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG,(%\"nG!\"\"%\"kG\"\"\"F)F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG%\"kG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:='f':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "RE:=subs(k=k+1,q)*f(k)-subs(k=k+1,r)*f(k-1)=p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,&*&,&%\"nG!\"\"%\"kG\"\"\"F,-%\"fG6#F+F,F,*& ,&F+F,F,F,F,-F.6#,&F+F,F,F*F,F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sol:=rsolve(RE,f(k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG,&*,,(*&-%\"fG6#\"\"!\"\"\"%\"nGF-F-F-F-F)F-F-,&F-F-F.F-! \"\"-%&GAMMAG6#,&F.F0F-F-F--F26#,(F.F0%\"kGF-F-F-F0-F26#,&F8F-\"\"#F-F -F-*&F-F-F/F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=findf (p,q,r,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,$*&\"\"\"F'%\"nG! \"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "s:=r/p*subs(k=k-1 ,f)*a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,$**%\"kG\"\"\"%\"nG! \"\")F*F'F(-%)binomialG6$F)F'F(F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Now we consider the particular case n=-10." }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a:=(-1)^k*binomial(-10,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG*&)!\"\"%\"kG\"\"\"-%)binomial G6$!#5F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rat:=subs(k=k +1,a)/a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ratG**)!\"\",&%\"kG\"\" \"F*F*F*-%)binomialG6$!#5F(F*)F'F)F'-F,6$F.F)F'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "rat:=normal(expand(rat));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ratG*&,&\"#5\"\"\"%\"kGF(F(,&F)F(F(F(!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "q:=numer(rat);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"qG,&\"#5\"\"\"%\"kGF'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "r:=denom(rat);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG,&%\"kG\"\"\"F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "q(k) and r(k+j) have a nontrivial gcd for j=9:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "gcd(q,subs(k=k+9,r));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"#5\"\"\" %\"kGF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "pqr:=update(p,su bs(k=k-1,q),subs(k=k-1,r),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pq rG7%*4,&%\"kG\"\"\"\"\"*F)F),&F(F)\"\")F)F),&F(F)\"\"(F)F),&F(F)\"\"'F )F),&F(F)\"\"&F)F),&F(F)\"\"%F)F),&F(F)\"\"$F)F),&F(F)\"\"#F)F),&F(F)F )F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p:=op(1,pqr); \+ q:=op(2,pqr); r:=op(3,pqr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG* 4,&%\"kG\"\"\"\"\"*F(F(,&F'F(\"\")F(F(,&F'F(\"\"(F(F(,&F'F(\"\"'F(F(,& F'F(\"\"&F(F(,&F'F(\"\"%F(F(,&F'F(\"\"$F(F(,&F'F(\"\"#F(F(,&F'F(F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:='f':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "RE:=subs(k=k+ 1,q)*f(k)-subs(k=k+1,r)*f(k-1)=p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#REG/,&-%\"fG6#%\"kG\"\"\"-F(6#,&F*F+F+!\"\"F/*4,&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+\"\"$F+F+,&F*F+\"\"#F+F+,&F*F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "sol:=rsolve(RE,f(k));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG,(-%\"fG6#\"\"!\"\"\"\"'!)GO!\"\"*8F+F*,&%\"kGF*F*F*F*,&* &\"\"#F,F/F*F*F*F*F*,&*&\"\"$F,F/F*F*F*F*F*,&*&\"\"%F,F/F*F*F*F*F*,&*& \"\"&F,F/F*F*F*F*F*,&*&\"\"'F,F/F*F*F*F*F*,&*&\"\"(F,F/F*F*F*F*F*,&*& \"\")F,F/F*F*F*F*F*,&*&\"\"*F,F/F*F*F*F*F*,&F*F**&\"#5F,F/F*F*F*F*" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=findf(p,q,r,k);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"fG,6*&\"(kG1\"\"\"\"%\"kGF(F(*&#\" ()ywj\"\"&F(*$)F)\"\"#F(F(F(*&\"']4%)F()F)\"\"$F(F(*&\"'$pT$F()F)\"\"% F(F(*&#\"'6/=F0F(*$)F)F-F(F(F(*&#\"'tx:\"#5F(*$)F)\"\"'F(F(F(*&\"%:=F( )F)\"\"(F(F(*&\"$K\"F()F)\"\")F(F(*&#\"#6F0F(*$)F)\"\"*F(F(F(*&#F(FAF( *$)F)FAF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "specials:= r/p*subs(k=k-1,f)*a;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)specialsG*: ,&%\"kG\"\"\"\"\"*F(!\"\",&F'F(\"\")F(F*,&F'F(\"\"(F(F*,&F'F(\"\"'F(F* ,&F'F(\"\"&F(F*,&F'F(\"\"%F(F*,&F'F(\"\"$F(F*,&F'F(\"\"#F(F*,&F'F(F(F( F*,8*&\"(kG1\"F(F'F(F(F " 0 "" {MPLTEXT 1 0 45 "difference:=simplify(specials-subs(n=-10,s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+differenceG,$*:\"'!)GO\"\"\")!\"\",&%\"kGF(F(F(F(-%) binomialG6$!#5F,F(,&F,F(\"\"*F(F*,&F,F(\"\")F(F*,&F,F(\"\"(F(F*,&F,F( \"\"'F(F*,&F,F(\"\"&F(F*,&F,F(\"\"%F(F*,&F,F(\"\"$F(F*,&F,F(\"\"#F(F*F +F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplify(differenc e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*:\"'!)GO\"\"\")!\"\",&%\"kGF &F&F&F&-%)binomialG6$!#5F*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(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "[seq(dif ference,k=1..10)]; k:='k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,!\"\"F $F$F$F$F$F$F$F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "third exampl e" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a:= binomial(n,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%)binomialG6$ %\"nG%\"kG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rat:=subs(k=k +1,a)/a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ratG*&-%)binomialG6$%\" nG,&%\"kG\"\"\"F,F,F,-F'6$F)F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rat:=normal(expand(rat));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ratG,$*&,&%\"nG!\"\"%\"kG\"\"\"F+,&F*F+F+F+F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "q:=numer(rat);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"qG,&%\"nG\"\"\"%\"kG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "r:=denom(rat);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG,&%\"kG\"\"\"F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "pqr:=update(p,subs(k=k- 1,q),subs(k=k-1,r),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pqrG7%\" \"\",(%\"nGF&%\"kG!\"\"F&F&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p:=op(1,pqr); q:=op(2,pqr); r:=op(3,pqr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG, (%\"nG\"\"\"%\"kG!\"\"F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG% \"kG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:='f':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "RE:=subs(k=k+1,q)*f(k)-subs(k=k+1,r )*f(k-1)=p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,&*&,&%\"nG\"\" \"%\"kG!\"\"F*-%\"fG6#F+F*F**&,&F+F*F*F*F*-F.6#,&F+F*F*F,F*F,F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rsolve(RE,f(k));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,&*,)!\"\"%\"kG\"\"\",,%\"nGF&\"\"#F&)F+,&F( F(F*F(F(*&-%\"fG6#\"\"!F(F*F(F(*&F/F()F*F+F(F(F(-%&GAMMAG6#,&F*F&F(F&F (-F66#,(F*F&F'F(F(F(F&-F66#,&F'F(F+F(F(F(*&-%*hypergeomG6%7$F(F;7#,&F' F(\"\"$F(F&F(F>F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=f indf(p,q,r,k);" }}{PARA 8 "" 1 "" {TEXT -1 41 "Error, (in findf) No po lynomial f exists\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "gosp er(a,k);" }}{PARA 8 "" 1 "" {TEXT -1 64 "Error, (in gosper) No hyperge ometric term antidifference exists\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a:='a': s:='s': p:='p': q:='q': r:='r': f:='f':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 22 "Zeilberger's Algorithm" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \"hsum9.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2004,~Wolfram~Koepf,~Univer sity~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sumrecur sion(k*binomial(n,k),k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*( \"\"#\"\"\",&F'F'%\"nGF'F'-%\"sG6#F)F'F'*&F)F'-F+6#F(F'!\"\"\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sumrecursion((-1)^k*binomial (n,k)^2,k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(\"\"%\"\"\",& F'F'%\"nGF'F'-%\"sG6#F)F'F'*&,&\"\"#F'F)F'F'-F+6#F.F'F'\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sumrecursion(binomial(n,k)^3 ,k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(\"\")\"\"\"),&F'F'% \"nGF'\"\"#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'F+F'-F-6#F;F'!\"\"\"\"!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 65 "With Zeilberger's algorithm, we can do more compli cated examples." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The Ap\351ry numbers" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "Sum(binomial(n,k)^2*binomial(n+k,k)^2,k=0..n); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)-%)binomialG6$%\"nG%\" kG\"\"#\"\"\")-F)6$,&F,F.F+F.F,F-F./F,;\"\"!F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "satisfy the recurrence equation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sumrecursion(binomial(n,k )^2*binomial(n+k,k)^2,k,A(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(* &),&\"\"\"F(%\"nGF(\"\"$F(-%\"AG6#F)F(F(*(,&F*F(*&\"\"#F(F)F(F(F(,(*& \"#F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "The power sums \+ of the binomial coefficients were worth a paper in the 1980s:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sumrecu rsion(binomial(n,k)^4,k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(* ,\"\"%\"\"\",&*&F&F'%\"nGF'F'\"\"&F'F',&\"\"$F'*&F&F'F*F'F'F',&F'F'F*F 'F'-%\"sG6#F*F'F'**\"\"#F',&F-F'*&F4F'F*F'F'F',(*&F-F')F*F4F'F'*&\"\"* F'F*F'F'\"\"(F'F'-F16#F/F'F'*&),&F4F'F*F'F-F'-F16#FAF'!\"\"\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sumrecursion(binomial(n,k)^5 ,k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,***\"#K\"\"\",(*&\"#bF' )%\"nG\"\"#F'F'*&\"$`#F'F,F'F'\"$#HF'F'),&F'F'F,F'\"\"%F'-%\"sG6#F,F'F '*&,0*&\"(qFo#F'F+F'F'*&\"'V0!*F')F,F3F'F'*&\"(kq#=F'F,F'F'*&\"&:%>F') F,\"\"'F'F'*&\"'*z0#F')F,\"\"&F'F'*&\"(t?3#F')F,\"\"$F'F'\"'[S^F'F'-F5 6#F2F'!\"\"*&,0*&\"%b6F'FBF'F'*&\"&`X\"F'FFF'F'*&\"'\\f?F'FJF'F'*&\"'F 3JF'F+F'F'*&\"&)\\vF'F=F'F'*&\"''eX#F'F,F'F'\"&?$zF'F'-F56#,&F-F'F,F'F 'FO*(,(*&F*F'F+F'F'*&\"$V\"F'F,F'F'\"#%*F'F'),&FKF'F,F'F3F'-F56#FcoF'F '\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sumrecursion(bino mial(n,k)^6,k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**0\"#C\"\" \",&*&\"\"'F'%\"nGF'F'\"\"&F'F',&\"\"$F'*&\"\"#F'F+F'F'F',&*&F*F'F+F'F '\"\"(F'F',**&\"#\"*F')F+F.F'F'*&\"$P'F')F+F0F'F'*&\"%\"\\\"F'F+F'F'\" %n6F'F'),&F'F'F+F'F.F'-%\"sG6#F+F'F'*&,6\")SV$H#F'*&\"*Ln\"z=F')F+F,F' F'*&\"*2=uy$F'F7F'F'*&\"*ou\" F')F+F3F'F'*&\")-g`rF')F+F*F'F'F'-FA6#F?F'!\"\"*(,&F0F'F+F'F',4*&\"%eM F'FWF'F'*&\"&dq&F'FgnF'F'*&\"'b&3%F'FjnF'F'*&\"(hnl\"F'FHF'F'*&\"(6#eT F'FOF'F'*&\"(a+h'F'F7F'F'*&\"(gl\\'F'F:F'F'*&\"(_#4OF'F+F'F'\"'S\"o)F' F'-FA6#F_oF'F]o**F_oF',**&F6F'F7F'F'*&\"$k$F'F:F'F'*&\"$!\\F'F+F'F'\"$ A#F'F'),&F.F'F+F'F,F'-FA6#F]qF'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sumrecursion(binomial(n,k)^7,k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,*,\"$G\"\"\"\",4*&\"'@xUF')%\"nG\"\")F'F'*&\"(! [w(*F')F,\"\"(F'F'*&\"):JP(*F')F,\"\"'F'F'*&\"*$)Q*=bF')F,\"\"&F'F'*& \"+9jqY>F')F,\"\"%F'F'*&\"+LpcvVF')F,\"\"$F'F'*&\"+eCp>hF')F,\"\"#F'F' *&\"+_@9p[F'F,F'F'\"+?\"*Q(o\"F'F'),&FEF'F,F'FEF'),&F'F'F,F'F5F'-%\"sG 6#F,F'F'*&,D*&\"3kCj,CK\"z9\"F'F@F'F'*&\"23]l)**33paF'FDF'F'\"1'p\")*p q!Q>#F'*&\"37&G#*Q\\Yiw\"F'F8F'F'*&\"3OF!\\h0sPm\"F'FF')F,\"#7F'F'*&\"1 ]3qsJ#G8\"F')F,\"#6F'F'*&\"3![GBX-u5U\"F'F4F'F'*&\"2pjER()z:k\"F')F,\" \"*F'F'*&\"2SaaE)*z5I%F'F+F'F'*&\"1ol^'=o4!\\F')F,\"#5F'F'*&\"2_vsu#G8 2;F'F,F'F'*&\"-g8#G1E\"F')F,\"#:F'F'*&\"+&yjUC$F')F,\"#;F'F'*&\"2%))*f I#o.U))F'F0F'F'F'-FN6#FJF'!\"\"*(,@*&\"07565A@H\"F'F4F'F'*&\"0;?(fReK? F'F8F'F'*&\"0GF'F@F'F'*&\"0KOgmEb7\"F'FDF'F'*&\"0[%Qs;poBF'Fo.$F'FgnF'F'*&\"+jl\"*)3\"F'F[oF'F'*&\",0@\">(z\"F'F_oF'F'*& \"-UnRz3=F'FcoF'F'*&\".()y&fixBr$F'F8F'F'*&\"2![S #y7k2#HF'F@F'F'*&\"2)))pqn8#o\\\"F'FDF'F'*&\"2Kv_5S[,)QF'F#QS]F'F[oF'F'\"0%)f1P(e7nF'*&\".#R?tv)Q(F'F_oF' F'*&\"/>/V7)\\g'F'FcoF'F'*&\"10?U-iWk " 0 "" {MPLTEXT 1 0 37 "sumr ecursion(binomial(n,k)^8,k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/ ,,*4\"#;\"\"\",&*&\"\")F'%\"nGF'F'\"#8F'F',&*&F*F'F+F'F'\"\"(F'F',&*&F *F'F+F'F'\"\"*F'F',&*&F*F'F+F'F'\"#6F'F',&\"\"#F'F+F'F',:*&\"*g`P-\"F' )F+F5F'F'*&\"+!3Lk=$F')F+\"#5F'F'*&\",=:hg\\%F')F+F2F'F'*&\"-2qD3'z$F' )F+F*F'F'*&\".]7+')38#F')F+F/F'F'*&\".A$H6+]$)F')F+\"\"'F'F'*&\"/#QYb& oIBF')F+\"\"&F'F'*&\"/d%y#G%Rj%F')F+\"\"%F'F'*&\"/er%eg:V'F')F+\"\"$F' F'*&\"/!4e[)oMfF')F+F7F'F'*&\"/_eaSywKF'F+F'F'\".?Id%4UDBF'FSF'F'*& \"-+k[30GF')F+\"#@F'F'*&\"0S?)Q(>?M$F')F+\"#>F'F'*&\"/+3)y[gS\"F')F+\" #?F'F'*&\"7B*)y#)\\Kh>>^KF'FEF'F'*&\"7P8$>?>%fy![3#F'FBF'F'*&\"66%*32J jYR2&[F'F;F'F'*&\"6-W4&3C-9\"fw\"F')F+F_oF'F'*&\"1S^YJ%4$4]F')F+\"#=F' F'*&\"4u`3lo)4=PEF')F+\"#:F'F'*&\"3F'F'*&\" 7*fV9[qKpD:;%F'FHF'F'*&\"7o`FkqerAsI6F'FWF'F'F'-F\\o6#F6F'!\"\"**F7F', H*&\"4Q))Q2XfJ2<%F'FKF'F'*&\"4Q(>+m#e#>DVF'FOF'F'*&\"4%)\\)zp(*GfA?F'F WF'F'*&\"3g&)4HuWc'G)F'FenF'F'*&\"4#=`&F'FjoF'F'\"2!o(=.b6X^#F'*&\"+![C:'pF'FhqF'F'*&\"2 ;lus.quo#F'FeqF'F'*&\"3>g3LOP@\"4\"F'F;F'F'*&\"3y_SYZ@I>\"*F'FBF'F'*& \"4kub7!R![Z*=F'FEF'F'*&\"3A>+K;\"QU_$F'F>F'F'*&\"-?*H!e1JF'FerF'F'*& \"3C%*f7djQ5@F'F+F'F'*&\"./h170]'F'F`rF'F'*&\"/g=0Q#HZ)F'F\\rF'F'*&\"4 ^J!R`jN2`JF'FHF'F'F'),&FXF'F+F'FXF'-F\\o6#FhuF'F_s**,:\",c,$eeaF'*&\"- 7yY*o]$F'F+F'F'*&\".mCY+x,\"F'FenF'F'*&\".!Q%f%egF'F'*&F:F'F;F'F'F 'FguF'),&FTF'F+F'F/F'-F\\o6#FdwF'F'**F*F'F6F',L*&\".!or)3H2(F'F[qF'F'* &\"0?()Re&GcJF'FgpF'F'*&\"1/^TV7m]mF'FhqF'F'*&\"23Q\"R#o?zz)F'FerF'F'* &\"3;Nbl6**R%>)F'F`rF'F'*&\"4/%R\"*G&R\"*>r&F'F\\rF'F'*&\"5(f?7lu@(z\" 4$F'F]pF'F'*&\"6F#4Q8QmFqI8F'FjoF'F'*&\"6SRaOSg\"p;DYF'FeqF'F'*&\"7SP9 #G#4&HD5J\"F'F;F'F'*&\"7TEw*y\"y:&4s/$F'F>F'F'*&\"7s^m.JrV\"**y\"eF'FB F'F'*&\"7r2b0pn+aa3\"*F'FEF'F'*&\"8c&=xdwdFF.j6F'FHF'F'*&\"8kU\\9D&pM \"[$*>\"F'FKF'F'*&\"7@t7`:r/%p`$)*F'FOF'F'*&\"7t\\k![W%yp\"RE'F'FSF'F' *&\"7[5M*[C,G*3')HF'FWF'F'*&\"7qX-(*G%*QiW-5F'FenF'F'*&\"6oA$)4cBN$=8@ F'F+F'F'\"5+YH`9!31R5#F'F'-F\\o6#FjnF'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Four different representations of the Legendre polynom ials:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "(a) W e consider the summand:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "legendre 1:=binomial(n,k)*binomial(-n-1,k)*((1-x)/2)^k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*legendre1G*(-%)binomialG6$%\"nG%\"kG\"\"\"-F'6$,&F)! \"\"F+F/F*F+),&#F+\"\"#F+*&F3F/%\"xGF+F/F*F+" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 7 "The sum" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Sum(legend re1,k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(-%)binomial G6$%\"nG%\"kG\"\"\"-F(6$,&F*!\"\"F,F0F+F,),&#F,\"\"#F,*&F4F0%\"xGF,F0F +F,/F+;\"\"!F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "has the hyperge ometric representation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Sumtohype r(legendre1,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*HypergeomG6%7$,$ %\"nG!\"\",&\"\"\"F+F(F+7#F+,&#F+\"\"#F+*&F/F)%\"xGF+F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "and satisfies the recurrence equation" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sumrecursion(legendre1,k,P(n));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&\"\"\"F'%\"nGF'F'-%\"PG6#F(F'F' *(%\"xGF',&*&\"\"#F'F(F'F'\"\"$F'F'-F*6#F&F'!\"\"*&,&F0F'F(F'F'-F*6#F6 F'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "(b) We consider the \+ summand:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "legendre2:=1/2^n*binomi al(n,k)^2*(x-1)^(n-k)*(x+1)^k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*l egendre2G**)\"\"#%\"nG!\"\"-%)binomialG6$F(%\"kGF'),&\"\"\"F)%\"xGF0,& F(F0F-F)F0),&F1F0F0F0F-F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "The s um" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Sum(legendre2,k=0..n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**)\"\"#%\"nG!\"\"-%)binomial G6$F)%\"kGF(),&\"\"\"F*%\"xGF1,&F)F1F.F*F1),&F2F1F1F1F.F1/F.;\"\"!F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "has the hypergeometric represen tation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Sumtohyper(legendre2,k); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*()\"\"#%\"nG!\"\"),&\"\"\"F'%\"xG F*F&F*-%*HypergeomG6%7$,$F&F'F07#F**&,&F+F*F*F*F*F)F'F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "and satisfies the recurrence equation" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sumrecursion(legendre2,k,P(n));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&\"\"\"F'%\"nGF'F'-%\"PG6#F(F'F' *(%\"xGF',&*&\"\"#F'F(F'F'\"\"$F'F'-F*6#F&F'!\"\"*&,&F0F'F(F'F'-F*6#F6 F'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "(c) We consider the \+ summand:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "legendre3:=1/2^n*(-1)^k *binomial(n,k)*binomial(2*n-2*k,n)*x^(n-2*k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*legendre3G*,)\"\"#%\"nG!\"\")F)%\"kG\"\"\"-%)binomia lG6$F(F+F,-F.6$,&*&F'F,F(F,F,*&F'F,F+F,F)F(F,)%\"xG,&F(F,*&F'F,F+F,F)F ," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "The sum" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Sum(legendre3,k=0..floor(n/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*,)\"\"#%\"nG!\"\")F*%\"kG\"\"\"-%)binomialG6$ F)F,F--F/6$,&*&F(F-F)F-F-*&F(F-F,F-F*F)F-)%\"xG,&F)F-*&F(F-F,F-F*F-/F, ;\"\"!-%&floorG6#,$*&F(F*F)F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "has the hypergeometric representation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Sumtohyper(legendre3,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,)\"\"#%\"nG!\"\"-%&GAMMAG6#,&*&F%\"\"\"F&F-F-F-F-F--F )6#,&F-F-F&F-!\"#)%\"xGF&F--%*HypergeomG6%7$,$*&F%F'F&F-F',&#F-F%F-*&F %F'F&F-F'7#,&F&F'F;F-*&F-F-*$)F3F%F-F'F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "and satisfies the recurrence equation" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "sumrecursion(legendre3,k,P(n));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,(*&,&\"\"\"F'%\"nGF'F'-%\"PG6#F(F'F'*(%\"xGF',&*&\" \"#F'F(F'F'\"\"$F'F'-F*6#F&F'!\"\"*&,&F0F'F(F'F'-F*6#F6F'F'\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "(d) We consider the summand:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "legendre4:=x^n*hyperterm([-n/2,(1-n )/2],[1],1-1/x^2,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*legendre4G* ,)%\"xG%\"nG\"\"\"-%+pochhammerG6$,$*&\"\"#!\"\"F(F)F0%\"kGF)-F+6$,&#F )F/F)*&F/F0F(F)F0F1F)-%*factorialG6#F1!\"#),&F)F)*&F)F)*$)F'F/F)F0F0F1 F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "The sum" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "Sum(legendre4,k=0..floor(n/2));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$SumG6$*,)%\"xG%\"nG\"\"\"-%+pochhammerG6$,$*&\"\"# !\"\"F)F*F1%\"kGF*-F,6$,&#F*F0F**&F0F1F)F*F1F2F*-%*factorialG6#F2!\"#) ,&F*F**&F*F**$)F(F0F*F1F1F2F*/F2;\"\"!-%&floorG6#,$*&F0F1F)F*F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "has the hypergeometric representat ion" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Sumtohyper(legendre4,k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"xG%\"nG\"\"\"-%*HypergeomG6%7$,$ *&\"\"#!\"\"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'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "and satisfies the recu rrence equation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sumrecursion(leg endre4,k,P(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&\"\"\"F'%\"n GF'F'-%\"PG6#F(F'F'*(%\"xGF',&*&\"\"#F'F(F'F'\"\"$F'F'-F*6#F&F'!\"\"*& ,&F0F'F(F'F'-F*6#F6F'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Proof of Clausen's formu la by Cauchy product:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "su mmand:=j->hyperterm([a,b],[a+b+1/2],1,j);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(summandGf*6#%\"jG6\"6$%)operatorG%&arrowGF(-%*hypert ermG6&7$%\"aG%\"bG7#,(F0\"\"\"F1F4#F4\"\"#F4F49$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Closedform(summand(j)*summand(k-j), j,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*HypertermG6&7%,$*&\"\"#\" \"\"%\"bGF*F*,$*&F)F*%\"aGF*F*,&F.F*F+F*7$,&*&F)F*F+F*F**&F)F*F.F*F*,( F.F*F+F*#F*F)F*F*%\"kG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Proof o f Clausen's formula by differential equations:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "The left hand factor satisfies the differential equa tion" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " DE:=sumdiffeq(summand(j)*x^j,j,C(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/,(**\"\"#\"\"\",&F)!\"\"%\"xGF)F)F,F)-%%diffG6$-%\"CG6#F,-% \"$G6$F,F(F)F)*&,.*(F(F)F,F)%\"aGF)F)F)F+*&F(F)%\"bGF)F+*&F(F)F9F)F+*( F(F)F,F)F;F)F)*&F(F)F,F)F)F)-F.6$F0F,F)F)**F(F)F0F)F9F)F;F)F)\"\"!" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Therefore the left hand side sati sfies the differential equation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "LHS:=`diffeq*diffeq`(DE,DE,C(x));" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%$LHSG,**&,&*(\"\")\"\"\"%\"aGF*)%\"bG\"\"#F*F**(F)F *)F+F.F*F-F*F*F*-%\"CG6#%\"xGF*F**&,8**\"#;F*F+F*F-F*F4F*F**(\"\"%F*F4 F*F,F*F**&F.F*F4F*F**(F:F*F4F*F0F*F**(\"\"'F*F4F*F+F*F**(F>F*F4F*F-F*F **&F:F*F,F*!\"\"*(F)F*F+F*F-F*FA*&F:F*F0F*FA*&F.F*F+F*FA*&F.F*F-F*FAF* -%%diffG6$F1F4F*F**&,.*(F>F*)F4F.F*F+F*F**(F>F*FLF*F-F*F**&F>F*FLF*F** &\"\"$F*F4F*FA*(F>F*F4F*F+F*FA*(F>F*F4F*F-F*FAF*-FG6$F1-%\"$G6$F4F.F*F **&,&*&F.F*FLF*FA*&F.F*)F4FPF*F*F*-FG6$F1-FV6$F4FPF*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "On the other hand the right hand side sat isfies the differential equation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "RHS:=sumdiffeq(hyperterm([2*a,2*b,a+b],[2 *a+2*b,a+b+1/2],x,k),k,C(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$RH SG/,**,\"\")\"\"\"-%\"CG6#%\"xGF)%\"aGF)%\"bGF),&F.F)F/F)F)F)*(\"\"#F) ,8*(F2F)F-F))F/F2F)F)*(F2F)F-F))F.F2F)F)**F(F)F.F)F/F)F-F)F)F-F)*&F2F) F5F)!\"\"*&F2F)F7F)F:F/F:F.F:*(\"\"$F)F-F)F/F)F)*(\"\"%F)F.F)F/F)F:*(F =F)F-F)F.F)F)F)-%%diffG6$F*F-F)F)**F=F)F-F),.*(F2F)F-F)F.F)F)F)F:*&F2F )F/F)F:*&F2F)F.F)F:*(F2F)F-F)F/F)F)*&F2F)F-F)F)F)-FB6$F*-%\"$G6$F-F2F) F)**F2F),&F)F:F-F)F))F-F2F)-FB6$F*-FN6$F-F=F)F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "These are equal:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "expand(LHS-op(1,RHS));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 46 "Differential Equations for Hypergeometric Sums" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The di fferential equation of the sine function:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sumdiffeq((-1)^k/(2*k+1)!*x^ (2*k+1),k,s(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"sG6#%\"xG\" \"\"-%%diffG6$F%-%\"$G6$F(\"\"#F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "The four different hypergeometric representations of the Legendre polynomials all lead to the same differential equation:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "legendre1:=binomial(n,k)*bin omial(-n-1,k)*((1-x)/2)^k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*legen dre1G*(-%)binomialG6$%\"nG%\"kG\"\"\"-F'6$,&F)!\"\"F+F/F*F+),&#F+\"\"# F+*&F3F/%\"xGF+F/F*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "su mdiffeq(legendre1,k,P(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(,&% \"xG\"\"\"F(F(F(,&F(!\"\"F'F(F(-%%diffG6$-%\"PG6#F'-%\"$G6$F'\"\"#F(F* *(F4F(F'F(-F,6$F.F'F(F**(F.F(%\"nGF(,&F(F(F9F(F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "legendre2:=1/2^n*binomial(n,k)^2*(x -1)^(n-k)*(x+1)^k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*legendre2G**) \"\"#%\"nG!\"\"-%)binomialG6$F(%\"kGF'),&\"\"\"F)%\"xGF0,&F(F0F-F)F0), &F1F0F0F0F-F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sumdiffeq( legendre2,k,P(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(,&%\"xG\"\" \"F(F(F(,&F(!\"\"F'F(F(-%%diffG6$-%\"PG6#F'-%\"$G6$F'\"\"#F(F**(F4F(F' F(-F,6$F.F'F(F**(F.F(%\"nGF(,&F(F(F9F(F(F(\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "legendre3:=1/2^n*(-1)^k*binomial(n,k)*binomial (2*n-2*k,n)*x^(n-2*k); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*legendre 3G*,)\"\"#%\"nG!\"\")F)%\"kG\"\"\"-%)binomialG6$F(F+F,-F.6$,&*&F'F,F(F ,F,*&F'F,F+F,F)F(F,)%\"xG,&F(F,*&F'F,F+F,F)F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sumdiffeq(legendre3,k,P(x));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,(*(,&%\"xG\"\"\"F(F(F(,&F(!\"\"F'F(F(-%%diffG6$-%\" PG6#F'-%\"$G6$F'\"\"#F(F**(F4F(F'F(-F,6$F.F'F(F**(F.F(%\"nGF(,&F(F(F9F (F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "legendre4:=x^ n*hyperterm([-n/2,(1-n)/2],[1],1-1/x^2,k); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*legendre4G*,)%\"xG%\"nG\"\"\"-%+pochhammerG6$,$*&\" \"#!\"\"F(F)F0%\"kGF)-F+6$,&#F)F/F)*&F/F0F(F)F0F1F)-%*factorialG6#F1! \"#),&F)F)*&F)F)*$)F'F/F)F0F0F1F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sumdiffeq(legendre4,k,P(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(,&%\"xG\"\"\"F(F(F(,&F(!\"\"F'F(F(-%%diffG6$-%\"PG 6#F'-%\"$G6$F'\"\"#F(F**(F4F(F'F(-F,6$F.F'F(F**(F.F(%\"nGF(,&F(F(F9F(F (F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 29 "A Generating Function Problem" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \"hsum9.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Maple~V~-~ Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2004,~Wol fram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "RE:=sumrecursion(binomial(alpha+n-1,n)*legendre4*z^n, n,s(k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,&*.)%\"zG\"\"#\"\" \",&F+!\"\"%\"xGF+F+,&F.F+F+F+F+,(*&F*F+%\"kGF+F+%&alphaGF+F+F+F+,&*&F *F+F2F+F+F3F+F+-%\"sG6#F2F+F+**\"\"%F+),&F2F+F+F+F*F+),&*&F.F+F)F+F+F+ F-F*F+-F76#F " 0 "" {MPLTEXT 1 0 21 "s ol:=rsolve(RE,s(k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG*4)*$)% \"zG\"\"#\"\"\"%\"kGF+),&F+!\"\"%\"xGF+F,F+),&F0F+F+F+F,F+)*&F+F+*$),& *&F0F+F)F+F+F+F/F*F+F/F,F+)\"\"%,$F,F/F+-%&GAMMAG6#,&*&F*F+F,F+F+%&alp haGF+F+-F=6#FAF/-F=6#,&F,F+F+F+!\"#-%\"sG6#\"\"!F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "We compute the initial value:" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "s(0)=Sum(binomial(alpha +n-1,n)*subs(k=0,legendre4)*z^n,n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"sG6#\"\"!-%$SumG6$*.-%)binomialG6$,(%&alphaG\"\"\" %\"nGF1F1!\"\"F2F1)%\"xGF2F1-%+pochhammerG6$,$*&\"\"#F3F2F1F3F'F1-F76$ ,&#F1F;F1*&F;F3F2F1F3F'F1-%*factorialGF&!\"#)%\"zGF2F1/F2;F'%)infinity G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "aw:=s(0)=sum(binomial( alpha+n-1,n)*subs(k=0,legendre4)*z^n,n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#awG/-%\"sG6#\"\"!*&\"\"\"F+),&F+F+*&%\"xGF+%\"zGF +!\"\"%&alphaGF1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Therefore we \+ get the solution:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sol:=subs(aw,sol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$solG*4)*$)%\"zG\"\"#\"\"\"%\"kGF+),&F+!\"\"%\"xGF+F,F+),&F0F+F+F+F ,F+)*&F+F+*$),&*&F0F+F)F+F+F+F/F*F+F/F,F+)\"\"%,$F,F/F+-%&GAMMAG6#,&*& F*F+F,F+F+%&alphaGF+F+-F=6#FAF/-F=6#,&F,F+F+F+!\"#),&F+F+F8F/FAF/" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "which we put into hypergeometric f orm:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " Sumtohyper(sol,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&),&\"\"\"F&*&% \"xGF&%\"zGF&!\"\"%&alphaGF*-%*HypergeomG6%7$,$*&\"\"#F*F+F&F&,&*&F2F* F+F&F&#F&F2F&7#F&**F)F2,&F&F*F(F&F&,&F(F&F&F&F&,&F'F&F&F*!\"#F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 24 "Combining the algorithms" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \"hsum9.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2004,~Wolfram~Koepf,~Univer sity~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "read \"F PS.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%OPackage~Formal~Power~Se ries,~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%TCopyrigh t~1995,~Dominik~Gruntz,~University~of~BaselG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%^oCopyright~2002,~Detlef~M|gzller~&~Wolfram~Koepf,~Uni versity~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "For" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Sum(x^( 3*k)/(3*k)!,k=0..infinity)=sum(x^(3*k)/(3*k)!,k=0..infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&)%\"xG,$*&\"\"$\"\"\"%\"kG F-F-F--%*factorialG6#F*!\"\"/F.;\"\"!%)infinityG-%*hypergeomG6%7\"7$#F -F,#\"\"#F,,$*&\"#FF2F)F,F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Ze ilberger's algorithm detects the differential equation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "DE:=sumdiffeq(x^(3*k )/(3*k)!,k,F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/,&-%\"FG6# %\"xG\"\"\"-%%diffG6$F'-%\"$G6$F*\"\"$!\"\"\"\"!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 69 "Maple's internal differential equation solver can \+ solve this equation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f:=rhs(dsolve(\{DE,F(0)=1,D(F)(0)=0,(D@@2)(F)(0)=0\}, F(x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*&#\"\"\"\"\"$F(-%$ expG6#%\"xGF(F(*&#\"\"#F)F(*&-F+6#,$*&F0!\"\"F-F(F6F(-%$cosG6#,$*(F0F6 F)#F(F0F-F(F(F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Reversely, the FPS algorithm redetects the differential equation from this repre sentation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "HolonomicDE(f,F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%di ffG6$-%\"FG6#%\"xG-%\"$G6$F+\"\"$\"\"\"F(!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "and recomputes the power series representation \+ of f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "FPS(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%*factorialG6#,$*&\"\"$\"\"\"%\" kGF-F-!\"\")%\"xGF*F-/F.;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 13 "Infinite \+ Sums" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \"hsum9.mpl\"; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summati on\",~Maple~V~-~Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyrigh t~1998-2004,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "read \"infhsum.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%fnThis~is~a~Maple~package~for~computing~recurrence~rel ations,G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%gnclosed~form~expressions ~and~uniformly~bounded~convergence~ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%gnnon-terminating~hypergeometric~series;~written~by~R.~VidunasG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q;~Version~4.25,~27-05-2002.6\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%LSupported~by~NWO,~project~number~61 3-06-565G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%SThe~help~function~is~en voked~by~\"~infhsumhelp(~)~\"G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Gauss identity" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "infclosedform(hyperterm([a,b],[c],1,k),k,c);" }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, The condition(s) for uniformly b ounded convergence are: 0 < Re(-a-b+c)\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**-%&GAMMAG6#,&%\"cG\"\"\"%\"aG!\"\"F+-F%6#,&F(F)%\"bGF+F+-F%6#F (F)-F%6#,(F*F+F/F+F(F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Kumme r's identity" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "infclosedform(hyperterm([a,b],[1+a-b],-1,k),k,a);" }}{PARA 7 " " 1 "" {TEXT -1 75 "Warning, The condition(s) for uniformly bounded co nvergence are: Re(b) < 0\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,)\"\"# ,$%\"aG!\"\"\"\"\"%#PiG#F)F%-%&GAMMAG6#,(F)F)F'F)%\"bGF(F)-F-6#,&*&F%F (F'F)F)F+F)F(-F-6#,(F)F)*&F%F(F'F)F)F0F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Pfaff-Saalsch\374tz identity" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "infclosedform(hyperterm([a,b ,c],[d,1+a+b+c-d],1,k),k,d);" }}{PARA 7 "" 1 "" {TEXT -1 79 "Warning, \+ The condition(s) for uniformly bounded convergence are: Re(a+b+c) < 1 \n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,&*2-%&GAMMAG6#,&%\"bG!\"\"%\"dG \"\"\"F*-F&6#,&%\"aGF*F+F,F*-F&6#,*F+F,F0F*F)F*%\"cGF*F*-F&6#,&F4F*F+F ,F*-F&6#,(F)F*F4F*F+F,F,-F&6#F+F,-F&6#,(F4F*F+F,F0F*F,-F&6#,(F+F,F0F*F )F*F,F,*6,*F)F,F0F,F4F,*&\"\"#F,F+F,F*F,-%*HypergeomG6%7',,*&FFF*F)F,F **&FFF*F0F,F**&FFF*F4F,F*F+F,F,F,F/F7F(F,7&,*F0F*F4F*F+F,F,F,,**&FFF*F )F,F**&FFF*F0F,F**&FFF*F4F,F*F+F,,*F+F,F,F,F0F*F)F*,*F)F*F4F*F+F,F,F,F *F,,(F0F,F4F,F+F*F*,(F+F*F0F,F)F,F*,(F)F,F4F,F+F*F*-F&6#F4F*-F&6#,,F,F ,F0F,F)F,F4F,F+F*F,F;F,-F&6#F)F*-F&6#F0F*F," }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 82 "Note that this is an non-obvious generalization of the \+ Pfaff-Saalsch\374tz identity. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 "Pe tkovsek's Algorithm" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \+ \"hsum9.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeo metric~Summation\",~Maple~V~-~Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2004,~Wolfram~Koepf,~University~of~KasselG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "For the following sum Zeilberger's algorithm finds a recurrence equation of order " }{TEXT 261 1 "c" } {TEXT -1 16 "-1 instead of 1:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Sum((-1)^k*binomial(n,k)*binomial(c*k,n),k=0. .n)=(-c)^k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*()!\"\"%\"kG \"\"\"-%)binomialG6$%\"nGF*F+-F-6$*&%\"cGF+F*F+F/F+/F*;\"\"!F/),$F3F)F *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "We compute:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "rec:=sumrecursion((- 1)^k*binomial(n,k)*binomial(4*k,n),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$recG/,**,\"#k\"\"\",&*&\"\"$F)%\"nGF)F)\"\"(F)F),&\" \"#F)F-F)F),&F)F)F-F)F)-%\"sG6#F-F)F)**\"\"%F),&*&F,F)F-F)F)F6F)F),(*& \"#PF))F-F0F)F)*&\"$!=F)F-F)F)\"$=#F)F)-F36#F/F)F)*,F,F)F*F)F7F),&*&F, F)F-F)F)\"\")F)F)-F36#,&F-F)F,F)F)F)**\"#;F)F/F),(*&\"#LF)F " 0 "" {MPLTEXT 1 0 46 "TIME:=time():\nrechyper(rec,s(n));\ntime()-TIME;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%K:!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Alternativel y, we load a package which includes an implementation of a much faster algorithm than Petkovsek's by Mark van Hoeij:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "TIME:=time():\n`LREtools/hso ls`(rec,s(n));\ntime()-TIME;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#)!\" %%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$\"H!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " }{TEXT 262 1 "c" }{TEXT -1 10 "=5, w e get" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "rec:=sumrecursion((-1)^k*binomial(n,k)*binomial(5*k,n),k,s(n));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%$recG/,,*2\"$D'\"\"\",&*&\"\"#F)%\"n GF)F)\"\"(F)F),&*&\"\"%F)F-F)F)\"#8F)F),&*&F1F)F-F)F)\"\"*F)F),&F-F)\" \"$F)F),&F,F)F-F)F),&F)F)F-F)F)-%\"sG6#F-F)F)*,\"#DF),&*&F1F)F-F)F)\" \"&F)F)F6F),,*&\"%[5F))F-F1F)F)*&\"&UA\"F))F-F7F)F)*&\"&>H&F))F-F,F)F) *&\"'z-5F)F-F)F)\"&-.(F)F)-F;6#F8F)F)*2\"\")F)F*F),&*&F,F)F-F)F)FAF)F) F/F)F3F)F?F),&*&F1F)F-F)F)\"#:F)F)-F;6#,&F-F)F1F)F)F)*0FAF),&*&F5F)F-F )F)\"#JF)F)F3F)F?F)FSF),(*&\"#TF)FKF)F)*&\"$$GF)F-F)F)\"$'[F)F)-F;6#F6 F)F)*.\"$D\"F)F/F)F6F)F8F),**&\"$_\"F)FHF)F)*&\"%)4\"F)FKF)F)*&\"%PCF) F-F)F)\"%B;F)F)-F;6#F9F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "# TIME:=time():\n# rechyper(rec,s(n));\n# time()-TIME;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "TIME:=time():\n`LREtools/hso ls`(rec,s(n));\ntime()-TIME;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#)!\" &%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$T%!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Wolfram Koepf: Hypergeometric Summation, Exerci se 9.3 (a):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "rec:=\nsumre cursion(hyperterm([-n,a,a+1/2,b],[2*a,(b-n+1)/2,(b-n)/2+1],1,k),k,s(n) );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$recG/,(*,,&\"\"\"F)%\"nGF)F), &%\"bGF)F*F)F),**&\"\"#F)%\"aGF)F)F)F)F,!\"\"F*F)F),(F,F1F*F)*&F/F)F0F )F)F)-%\"sG6#F*F)F)*.F/F),(F,F)F*F)F)F)F),&F,F)F*F1F),(F0F)F)F)F*F)F)F -F)-F56#F(F)F)*,,(F,F)F*F)F/F)F)F9F),(F,F)F*F1F)F1F),(*&F/F)F0F)F)F)F) F*F)F)-F56#,&F/F)F*F)F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "TIME:=time():\nres2:=`LREtools/hsols`(rec,s(n));\ntime()-TIME; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%res2G7$*,-%&GAMMAG6#,(%\"bG!\" \"%\"nG\"\"\"*&\"\"#F.%\"aGF.F.F.-F(6#,&F.F.F-F.F.-F(6#,&F-F.F+F,F,-F( 6#,&F-F.*&F0F.F1F.F.F,,&F+F.F-F.F,*(F'F.F5F,F " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Hyperexponential Integrati on" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \"hsum9.mpl\";" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation \",~Maple~V~-~Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~ 1998-2004,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Continuous version of Gosper's algorithm." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Does the func tion" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " f:=exp(x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%$expG6#*$)%\"x G\"\"#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "have a hyperexpon ential antiderivative? The answer is" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "contgosper(exp(x^2),x);" }}{PARA 8 "" 1 "" {TEXT -1 65 "Error, (in contgosper) No hyperexponential antide rivative exists\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The situatio n is different for" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "contgosper(x*exp(x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&-%$expG6#*$)%\"xGF'F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Let's do a more complicated example:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "term:=d iff((1+x^2)/(1-x^10),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%termG,& *(\"\"#\"\"\"%\"xGF(,&F(F(*$)F)\"#5F(!\"\"F.F(**F-F(,&F(F(*$)F)F'F(F(F (F*!\"#F)\"\"*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "res:=co ntgosper(term,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG,$*(,&\"\" \"F(*$)%\"xG\"\"#F(F(F(,**$)F+\"\"'F(F(*$)F+\"\"&F(!\"\"F+F(F(F4F4,,*$ )F+\"\"%F(F(*$)F+\"\"$F(F(F)F(F+F(F(F(F4F4" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "res:=normal(res);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$resG,$*(,&\"\"\"F(*$)%\"xG\"\"#F(F(F(,**$)F+\"\"'F(F(*$)F+\"\"&F( !\"\"F+F(F(F4F4,,*$)F+\"\"%F(F(*$)F+\"\"$F(F(F)F(F+F(F(F(F4F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "res:=normal(res,expanded);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG*&,&\"\"\"!\"\"*$)%\"xG\"\"#F 'F(F',&F'F(*$)F+\"#5F'F'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Let 's check Maple's internal integrator: " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "res:=int(term,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$resG,F*&#\"\"#\"\"&\"\"\"*&,&\"#5F**&F(F*F)#F*F(! \"\"#F0F(-%'arctanG6#*&,(*&\"\"%F*%\"xGF*F*F*F**$F)F/F*F*F,F1F*F*F0**F )F0,**&,&*&\"\")F*F)F/F0*&,&F:F*F)F0F*,&F:F*F*F0F*F0F*F9F*F**(F(F*F)F/ FCF*F0*&F8F*F)F/F0\"#?F*F*,&F-F**&F(F*F)F/F*F0,**&F(F*)F9F(F*F*F9F0*&F )F/F9F*F*F(F*F0F**&#F8F)F**(FG#!\"$F(-F36#*&,(*&F8F*F9F*F*F*F*F:F0F*FG F1F*F)F/F*F**&#F(F)F**&F,F1-F36#*&,(*&F8F*F9F*F*F*F0F:F0F*F,F1F*F*F**( F8F*F,FPFZF*F0*(F8F*F,FPF2F*F**&FXF**&FGF1-F36#*&,(*&F8F*F9F*F*F*F0F:F *F*FGF1F*F*F**&#F8F)F**(F,FPF2F*F)F/F*F0**F)F0,**&,&*&F@F*F)F/F**&FBF* ,&F:F0F*F*F*F0F*F9F*F**(F(F*F)F/F[pF*F**&F8F*F)F/F0FFF*F*FGF0,**&F(F*F KF*F*F9F*FLF0F(F*F0F**&#F(F)F**&FGF1FRF*F*F0*&FNF**(F,FPFZF*F)F/F*F**& #F8F)F**(FGFPF]oF*F)F/F*F0*&F*F**&F)F*,&F9F*F*F*F*F0F**(F8F*FGFPFRF*F* *&F*F**&F)F*,&F*F0F9F*F*F0F0**F)F0,**&,&*&F@F*F)F/F0*&,&F:F0F)F0F*,&F: F*F*F*F*F0F*F9F*F**(F(F*F)F/FfqF*F0*&F8F*F)F/F*FFF*F*F,F0,**&F(F*FKF*F *F9F*FLF*F(F*F0F**(F8F*FGFPF]oF*F0**F)F0,**&,&*&F@F*F)F/F**&FeqF*,&F:F 0F*F0F*F0F*F9F*F**(F(F*F)F/FbrF*F**&F8F*F)F/F*FFF*F*F,F0,**&F(F*FKF*F* F9F0FLF0F(F*F0F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "res:=no rmal(res);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$resG,$*6\"$?$\"\"\",& F(F(*$)%\"xG\"\"#F(F(F(,&\"\"&F(*$F/#F(F-F(!\"\",**&F-F(F+F(F(F,F2*&F/ F1F,F(F(F-F(F2,**&F-F(F+F(F(F,F(F5F2F-F(F2,&F,F(F(F(F2,&F(F2F,F(F2,&F0 F(F/F2F2,**&F-F(F+F(F(F,F(F5F(F-F(F2,**&F-F(F+F(F(F,F2F5F2F-F(F2F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "res:=normal(res,expanded); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG*&,&\"\"\"!\"\"*$)%\"xG\"\" #F'F(F',&F'F(*$)F+\"#5F'F'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "L et's check Risch's algorithm:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "`int/risch`(term,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\",&#F&F%!\"\"*&F%F)%\"xGF%F)F&,&F&F)*$)F+ \"#5F&F&F)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 51 "Differential and Recurrence Equations f or Integrals" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \"hsum9 .mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~ Summation\",~Maple~V~-~Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YC opyright~1998-2004,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "We would like to compute:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Int(x^2/(x^4+t^2)/(1 +t^2),t=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(% \"xG\"\"#,&*$)F'\"\"%\"\"\"F-*$)%\"tGF(F-F-!\"\",&F-F-F.F-F1/F0;\"\"!% )infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "integrand:=x^ 2/(x^4+t^2)/(1+t^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*integrandG* (%\"xG\"\"#,&*$)F&\"\"%\"\"\"F,*$)%\"tGF'F,F,!\"\",&F,F,F-F,F0" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "The integrand is a hyperexponentia l term:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "contratio(integrand,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\" \"#\"\"\"%\"tGF&,(F&F&*&F%F&)F'F%F&F&*$)%\"xG\"\"%F&F&F&,&F&F&*$F*F&F& !\"\",&F+F&F0F&F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "What type \+ of result should we expect?" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "[ratio(integrand,x),contratio(integrand,x)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$**,&%\"xG\"\"\"F'F'\"\"#,&*$)F&\" \"%F'F'*$)%\"tGF(F'F'F',.F*F'*&F,F')F&\"\"$F'F'*&\"\"'F')F&F(F'F'*&F,F 'F&F'F'F'F'F-F'!\"\"F&!\"#,$*,F(F',&F/F8*$F6F'F'F',&F=F'F/F'F'F&F8F)F8 F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Application of the continuo us version of Zeilberger's algorithm:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "RE:=intrecursion(integrand,t,S(x)); " }}{PARA 8 "" 1 "" {TEXT -1 78 "Error, (in intrecursion) Algorithm fi nds no recurrence equation of order <= 5\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "DE:=intdiffeq(integrand,t,S(x));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#DEG/,(*,,&\"\"\"!\"\"%\"xGF)F),&F+F)F)F)F),&F)F)*$ )F+\"\"#F)F)F)-%%diffG6$-%\"SG6#F+-%\"$G6$F+F0F)F+F)F)*&,&F)F)*&\"\"(F ))F+\"\"%F)F)F)-F26$F4F+F)F)*(\"\")F)F4F))F+\"\"$F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(DE,S(x));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"SG6#%\"xG,&*&%$_C1G\"\"\",&*$)F'\"\"%F+F+F+!\"\" F0F+*(%$_C2GF+F'\"\"#F,F0F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "res:=int(integrand,t=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG,$*&#\"\"\"\"\"#F(*(%#PiGF(,&-%%csgnG6#*$)-%*conjugateG6# %\"xGF)F(!\"\"*$)F5F)F(F(F(,&*$)F5\"\"%F(F(F(F6F6F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(x>0):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "res:=normal(res);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG,$*(\"\"#!\"\"%#PiG\"\"\",&*$)%#x|irGF'F*F*F*F*F (F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Which recurrence equation \+ is valid for the result S(x)?" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ratio(res,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$)%#x|irG\"\"#\"\"\"F)F)F)F),(F%F)*&F(F)F'F)F)F(F)!\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rat:=factor(ratio(res,x),I) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ratG**,&%#x|irG\"\"\"^#!\"\"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 41 "Hence the recurrence equation for S(x) is" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "denom(r at)*S(x+1)-numer(rat)*S(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*( ,&%#x|irG!\"\"^$F(\"\"\"F*F*,&F'F*^$F*F*F*F*-%\"SG6#,&F'F*F*F*F*F**(,& F'F(^#F*F*F*,&F'F*F3F*F*-F.6#F'F*F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x:='x':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Rodrigues Formulas" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \"hsum9.mpl\";" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\" ,~Maple~V~-~Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~19 98-2004,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Rodrigues formula of the Legendre polynomials" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "P(n,x)= (-1)^n/2^n/n!*diff((1-x^2)^n,x$n); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"PG6$%\"nG%\"xG**)!\"\"F'\"\"\")\"\"#F'F+-%*factorialG6#F'F+-%%di ffG6$),&F,F,*$)F(F.F,F+F'-%\"$G6$F(F'F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "The following function computes the recurrence equation o f the family by Cauchy's integral formula " }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "RE:=rodriguesrecursion((-1)^ n/2^n/n!,(1-x^2)^n,x,P(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/ ,(*&,&\"\"#\"\"\"%\"nGF*F*-%\"PG6#F(F*F**(%\"xGF*,&\"\"$F**&F)F*F+F*F* F*-F-6#,&F*F*F+F*F*!\"\"*&F6F*-F-6#F+F*F*\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Similarly, we get the differential equation" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DE:=rod riguesdiffeq((-1)^n/2^n/n!,(1-x^2)^n,n,P(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/,(*(,&\"\"\"!\"\"%\"xGF)F),&F+F)F)F)F)-%%diffG6$ -%\"PG6#F+-%\"$G6$F+\"\"#F)F**(F6F)F+F)-F.6$F0F+F)F**(F0F)%\"nGF),&F)F )F;F)F)F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "The holonomic \+ recurrence equation defines the Legendre polynomials uniquely up to th e initial values" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "P(0,x)= eval(subs(n=0,(-1)^n/2^n/n!*(1-x^2)^n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6$\"\"!%\"xG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " P(1,x)=eval(subs(n=1,(-1)^n/2^n/n!*diff((1-x^2)^n,x$n)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6$\"\"\"%\"xGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Rodrigues formula of the generalized Laguerre polynomi als" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "L (n,alpha,x)=exp(x)/n!/x^alpha*diff(exp(-x)*x^(alpha+n),x$n); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"LG6%%\"nG%&alphaG%\"xG**-%$expG6# F)\"\"\"-%*factorialG6#F'!\"\")F)F(F2-%%diffG6$*&-F,6#,$F)F2F.)F),&F(F .F'F.F.-%\"$G6$F)F'F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "The foll owing function computes the recurrence equation of the family by Cauch y's integral formula " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "RE:=rodriguesrecursion(exp(x)/n!/x^alpha,exp(-x)*x^(a lpha+n),x,L(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,(*&,&\"\"# \"\"\"%\"nGF*F*-%\"LG6#F(F*F**&,*%&alphaG!\"\"\"\"$F2*&F)F*F+F*F2%\"xG F*F*-F-6#,&F*F*F+F*F*F**&,(F1F*F+F*F*F*F*-F-6#F+F*F*\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Similarly, we get the differential equati on" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "DE :=rodriguesdiffeq(exp(x)/n!/x^alpha,exp(-x)*x^(alpha+n),n,L(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/,(*&%\"xG\"\"\"-%%diffG6$-%\"LG 6#F(-%\"$G6$F(\"\"#F)F)*&,(F(F)%&alphaG!\"\"F)F7F)-F+6$F-F(F)F7*&F-F)% \"nGF)F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "The holonomic r ecurrence equation defines the Legendre polynomials uniquely up to the initial values" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "L(0,alph a,x)=simplify(subs(n=0,exp(x)/n!/x^alpha*exp(-x)*x^(alpha+n)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"LG6%\"\"!%&alphaG%\"xG\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "L(1,alpha,x)=simplify(subs(n=1,exp( x)/n!/x^alpha*diff((exp(-x)*x^(alpha+n),x$n))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"LG6%\"\"\"%&alphaG%\"xG,(F)!\"\"F(F'F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 20 "Generating Functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \"hsum9.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2004,~Wolfram~Koepf,~Univer sity~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "The generating function of the generalized Laguerre polynomials satisfies the recurr ence equation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "GFrecursion((1-z)^(-alpha-1)*exp((x*z)/(z-1)),1,z,L(n ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&\"\"#\"\"\"%\"nGF(F(-%\" LG6#F&F(F(*&,*%&alphaG!\"\"\"\"$F0*&F'F(F)F(F0%\"xGF(F(-F+6#,&F(F(F)F( F(F(*&,(F/F(F)F(F(F(F(-F+6#F)F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "compare:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "RE;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&\"\"#\" \"\"%\"nGF(F(-%\"LG6#F&F(F(*&,*%&alphaG!\"\"\"\"$F0*&F'F(F)F(F0%\"xGF( F(-F+6#,&F(F(F)F(F(F(*&,(F/F(F)F(F(F(F(-F+6#F)F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "and the differential equation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "GFdiffeq((1-z)^( -alpha-1)*exp((x*z)/(z-1)),1,z,n,L(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"xG\"\"\"-%%diffG6$-%\"LG6#F&-%\"$G6$F&\"\"#F'F'*&,(F&F'% &alphaG!\"\"F'F5F'-F)6$F+F&F'F5*&F+F'%\"nGF'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "compare:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 3 "DE;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&% \"xG\"\"\"-%%diffG6$-%\"LG6#F&-%\"$G6$F&\"\"#F'F'*&,(F&F'%&alphaG!\"\" F'F5F'-F)6$F+F&F'F5*&F+F'%\"nGF'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The initial values:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "series((1-z)^(-alpha-1)*exp((x*z)/(z-1)), z=0,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"zG\"\"\"\"\"!,(F%F%%&a lphaGF%%\"xG!\"\"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--%\"OG6#F%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }