{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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 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 "Courier" 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 "Headi ng 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 "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 "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 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 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 "Normal" -1 258 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 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Wolfram Koepf " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 51 "Computer Al gebra Methods for Orthogonal Polynomials" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 15 "Maple Worksheet" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }{TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 49 "Conversion of Recurrence and Difference E quations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(LREtools); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7:%6AnalyticityConditionsG%3Hyperg eometricTermG%3IsDesingularizableG%*REcontentG%)REcreateG%'REplotG%+RE primpartG%.REreduceorderG%'REtoDEG%*REtodeltaG%)REtoprocG%.ValuesAtPoi ntG%/autodispersionG%.constcoeffsolG%1dAlembertiansolsG%&deltaG%+dispe rsionG%(divconqG%)firstlinG%.hypergeomsolsG%)polysolsG%,ratpolysolsG%( riccatiG%&shiftG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "RE:=n*f (n+2)-(n-1)*(n+1)*f(n+1)+f(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R EG,(*&%\"nG\"\"\"-%\"fG6#,&F'F(\"\"#F(F(F(*(,&F'F(F(!\"\"F(,&F'F(F(F(F (-F*6#F1F(F0-F*6#F'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Conversi on to a difference equation:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "deltaexpr:=REtodelta(RE,f(n),\{\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*deltaexprG,,*&%\"nG\"\"\")&&%)LREtoolsG6# %&DeltaG6#F'\"\"#F(F(*&,(*$)F'F0F(!\"\"F(F(*&F0F(F'F(F(F(F*F(F(F'F(F0F (F3F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "subs(LREtools[Delt a][n]=Delta,deltaexpr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&%\"nG\" \"\")%&DeltaG\"\"#F&F&*&,(*$)F%F)F&!\"\"F&F&*&F)F&F%F&F&F&F(F&F&F%F&F) F&F,F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Now we convert back" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "read \" deltatore.mpl\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "deltato RE(deltaexpr,f(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"nG\"\"\" -%\"fG6#,&F%F&\"\"#F&F&F&*(,&F%F&F&!\"\"F&,&F%F&F&F&F&-F(6#F/F&F.-F(6# F%F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "and compare with the orig inal equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "RE;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"nG\"\"\"-%\"fG6#,&F%F&\"\"#F&F& F&*(,&F%F&F&!\"\"F&,&F%F&F&F&F&-F(6#F/F&F.-F(6#F%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 49 "Coefficients of Solution of Differential Equation" }}{EXCHG {PARA 0 " " 0 "" {TEXT 261 25 "We define the polynomials" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sigma;" "6#%&sigmaG" }{TEXT -1 1 " " }{TEXT 262 3 "and " }{TEXT -1 1 " " }{XPPEDIT 18 0 "tau;" "6#%$tauG" }{TEXT -1 1 " " } {TEXT 263 38 "with arbitrary coefficients a,b,c,d,e:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sigma:=a*x^2+b*x+c;\nt au:=d*x+e;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG,(*&%\"aG\"\"\" )%\"xG\"\"#F(F(*&%\"bGF(F*F(F(%\"cGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tauG,&*&%\"dG\"\"\"%\"xGF(F(%\"eGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "and consider the differential equation" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "DE:=sigma*diff(F(x),x$2 )+tau*diff(F(x),x)-n*(a*n+d-a)*F(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#DEG,(*&,(*&%\"aG\"\"\")%\"xG\"\"#F*F**&%\"bGF*F,F*F*%\"cGF*F*-%%d iffG6$-%\"FG6#F,-%\"$G6$F,F-F*F**&,&*&%\"dGF*F,F*F*%\"eGF*F*-F26$F4F,F *F**(%\"nGF*,(*&F)F*FBF*F*F=F*F)!\"\"F*F4F*FE" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 116 "To convert the differential equation to a recurrenc e equation for the series coefficients, we load the gfun package." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gf un):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "RE:=`diffeqtorec`(D E,F(x),A(j));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG,(*&,,*&%\"aG\" \"\")%\"jG\"\"#F*F**&,&%\"dGF*F)!\"\"F*F,F*F**&)%\"nGF-F*F)F*F1*&F4F*F 0F*F1*&F)F*F4F*F*F*-%\"AG6#F,F*F**&,(*&%\"bGF*F+F*F**&,&%\"eGF*F=F*F*F ,F*F*F@F*F*-F86#,&F,F*F*F*F*F**&,(*&%\"cGF*F+F*F**(\"\"$F*FGF*F,F*F**& F-F*FGF*F*F*-F86#,&F,F*F-F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "map(factor,RE);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(,&%\"jG !\"\"%\"nG\"\"\"F),**&%\"aGF)F(F)F)%\"dGF)F,F'*&F,F)F&F)F)F)-%\"AG6#F& F)F'*(,&F&F)F)F)F),&*&%\"bGF)F&F)F)%\"eGF)F)-F06#F3F)F)**%\"cGF)F3F),& F&F)\"\"#F)F)-F06#F " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 "Computing the Recurrence Co efficients" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 89 "Continuous case: W e consider the three highest coefficients of the orthogonal polynomial :" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "p:= k[n]*x^n+kprime[n]*x^(n-1)+kprimeprime[n]*x^(n-2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"pG,(*&&%\"kG6#%\"nG\"\"\")%\"xGF*F+F+*&&%'kprimeG F)F+)F-,&F*F+F+!\"\"F+F+*&&%,kprimeprimeGF)F+)F-,&F*F+\"\"#F3F+F+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 257 61 "The polynomial satisfies the diff erential equation DE=0 with:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DE:=sigma*diff(p,x$2)+tau*diff(p,x)+lambda[n]*p;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#DEG,(*&,(*&%\"aG\"\"\")%\"xG\"\"#F*F**&%\"bGF*F ,F*F*%\"cGF*F*,.**&%\"kG6#%\"nGF*)F,F6F*F6F-F,!\"#F***F3F*F7F*F6F*F,F8 !\"\"**&%'kprimeGF5F*)F,,&F6F*F*F:F*F?F-F,F8F***FF*F?F*F,F8F:**&% ,kprimeprimeGF5F*)F,,&F6F*F-F:F*FEF-F,F8F***FBF*FDF*FEF*F,F8F:F*F**&,& *&%\"dGF*F,F*F*%\"eGF*F*,(**F3F*F7F*F6F*F,F:F***FF*F?F*F,F:F***FB F*FDF*FEF*F,F:F*F*F**&&%'lambdaGF5F*,(*&F3F*F7F*F**&FF*F**&FBF*FD F*F*F*F*" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 24 "We collect coeffici ents:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "de:=collect(simplify(DE/x^(n-4)),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#deG,0*&,**(%\"aG\"\"\"&%\"kG6#%\"nGF*F.F*!\"\"*&&%'lambdaGF-F*F+ F*F**(%\"dGF*F+F*F.F*F**(F)F*F+F*)F.\"\"#F*F*F*)%\"xG\"\"%F*F**&,4**\" \"$F*F)F*&%'kprimeGF-F*F.F*F/*(%\"bGF*F+F*F6F*F**(F)F*F?F*F6F*F**(F7F* F)F*F?F*F**&F1F*F?F*F**&F4F*F?F*F/*(%\"eGF*F+F*F.F*F**(FBF*F+F*F.F*F/* (F4F*F?F*F.F*F*F*)F9F>F*F**&,<**\"\"&F*F)F*&%,kprimeprimeGF-F*F.F*F/*( F7F*F4F*FPF*F/*&FHF*F?F*F/*(%\"cGF*F+F*F.F*F/**F>F*FBF*F?F*F.F*F/*(F7F *FBF*F?F*F**(FUF*F+F*F6F*F**(FHF*F?F*F.F*F**(\"\"'F*F)F*FPF*F**(F4F*FP F*F.F*F**&F1F*FPF*F**(F)F*FPF*F6F*F**(FBF*F?F*F6F*F*F*)F9F7F*F**&,2*(F UF*F?F*F6F*F***FOF*FBF*FPF*F.F*F/*(F7F*FUF*F?F*F***F>F*FUF*F?F*F.F*F/* (FHF*FPF*F.F*F**(FBF*FPF*F6F*F**(FenF*FBF*FPF*F**(F7F*FHF*FPF*F/F*F9F* F***FOF*FUF*FPF*F.F*F/*(FenF*FUF*FPF*F**(FUF*FPF*F6F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 73 "Equating the highest coefficient gives t he already mentioned identity for" }{TEXT -1 1 " " }{XPPEDIT 18 0 "lam bda;" "6#%'lambdaG" }{TEXT -1 1 ":" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "rule1:=lambda[n]=solve(coeff(de,x,4 ),lambda[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule1G/&%'lambdaG6 #%\"nG,$*&F)\"\"\",(*&%\"aGF,F)F,F,%\"dGF,F/!\"\"F,F1" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 24 "This can be substituted:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "de:=expand(subs(ru le1,de));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#deG,X**\"\"#\"\"\")%\" xG\"\"$F(%\"aGF(&%'kprimeG6#%\"nGF(F(**\"\"'F()F*F'F(F,F(&%,kprimeprim eGF/F(F(**F'F(F3F(%\"bGF(F-F(F(**F2F(F*F(F7F(F4F(F(**F'F(F*F(%\"cGF(F- F(F(*(F2F(F:F(F4F(F(*(F)F(%\"dGF(F-F(!\"\"**F'F(F3F(F=F(F4F(F>*(F3F(% \"eGF(F-F(F>**F'F(F*F(FAF(F4F(F>**F3F(F:F(&%\"kGF/F()F0F'F(F(**F3F(F:F (FDF(F0F(F>*,F'F(F)F(F,F(F-F(F0F(F>*,\"\"%F(F3F(F,F(F4F(F0F(F>**F)F(F7 F(FDF(FFF(F(**F)F(F7F(FDF(F0F(F>**F3F(F7F(F-F(FFF(F(*,F+F(F3F(F7F(F-F( F0F(F>**F*F(F7F(F4F(FFF(F(*,\"\"&F(F*F(F7F(F4F(F0F(F>**F*F(F:F(F-F(FFF (F(*,F+F(F*F(F:F(F-F(F0F(F>*(F:F(F4F(FFF(F(**FQF(F:F(F4F(F0F(F>**F)F(F AF(FDF(F0F(F(**F3F(FAF(F-F(F0F(F(**F*F(FAF(F4F(F0F(F(" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 81 "Equating the second highest coefficient gives k'[n] as rational multiple of k[n]:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "rule2:=kprime[n]=solve(coeff(de,x,3),kprime[n]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule2G/&%'kprimeG6#%\"nG**&%\"kGF( \"\"\"F)F-,(%\"eGF-*&%\"bGF-F)F-F-F1!\"\"F-,(*&\"\"#F-%\"aGF-F2%\"dGF- *(F5F-F6F-F)F-F-F2" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 81 "Equating \+ the third highest coefficient gives k''[n] as rational multiple of k[n ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "rule3:=kprimeprime[n] =solve(coeff(subs(rule2,de),x,2),kprimeprime[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule3G/&%,kprimeprimeG6#%\"nG,$*&#\"\"\"\"\"#F-*,&% \"kGF(F-F)F-,>*(\"\"$F-%\"bGF-%\"eGF-F-*(\"\"&F-)F5F.F-F)F-F-*&F.F-F9F -!\"\"*&)F6F.F-F)F-F-**F.F-F6F-)F)F.F-F5F-F-**F8F-F6F-F)F-F5F-F;*$F=F- F;**\"\"%F-%\"cGF-F)F-%\"aGF-F;*(FDF-F)F-%\"dGF-F-**F.F-FDF-F?F-FEF-F- *(F.F-FDF-FEF-F-*&FDF-FGF-F;*&F9F-)F)F4F-F-*(FCF-F9F-F?F-F;F-,(*&F.F-F EF-F;FGF-*(F.F-FEF-F)F-F-F;,(*&F4F-FEF-F;FGF-*(F.F-FEF-F)F-F-F;F-F-" } }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 60 "Without loss of generality we \+ consider the monic case, hence" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "k[n]:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%\"kG6#%\"nG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 13 "and there fore" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "rule2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%'kprimeG6#%\"nG*(F'\"\"\",(%\"eGF)*&%\"bGF)F' F)F)F-!\"\"F),(*&\"\"#F)%\"aGF)F.%\"dGF)*(F1F)F2F)F'F)F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "rule3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%,kprimeprimeG6#%\"nG,$*,\"\"#!\"\"F'\"\"\",>*(\"\"$F,%\"bGF,% \"eGF,F,*(\"\"&F,)F0F*F,F'F,F,*&F*F,F4F,F+*&)F1F*F,F'F,F,**F*F,F1F,)F' F*F,F0F,F,**F3F,F1F,F'F,F0F,F+*$F7F,F+**\"\"%F,%\"cGF,F'F,%\"aGF,F+*(F >F,F'F,%\"dGF,F,**F*F,F>F,F9F,F?F,F,*(F*F,F>F,F?F,F,*&F>F,FAF,F+*&F4F, )F'F/F,F,*(F=F,F4F,F9F,F+F,,(*&F*F,F?F,F+FAF,*(F*F,F?F,F'F,F,F+,(*&F/F ,F?F,F+FAF,*(F*F,F?F,F'F,F,F+F," }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 94 "We would like to compute the coefficients a(n), b(n) and c(n) in t he recurrence equation RE=0:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 47 "RE:=x*P(n)-(a[n]*P(n+1)+b[n]*P(n)+c[n]*P(n-1)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG,**&%\"xG\"\"\"-%\"PG6#%\"n GF(F(*&&%\"aGF+F(-F*6#,&F,F(F(F(F(!\"\"*&&%\"bGF+F(F)F(F3*&&%\"cGF+F(- F*6#,&F,F(F(F3F(F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "RE:=s ubs(\{P(n)=p,P(n+1)=subs(n=n+1,p),P(n-1)=subs(n=n-1,p)\},RE);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#REG,**&%\"xG\"\"\",()F'%\"nGF(*&&%' kprimeG6#F+F()F',&F+F(F(!\"\"F(F(*&&%,kprimeprimeGF/F()F',&F+F(\"\"#F2 F(F(F(F(*&&%\"aGF/F(,()F',&F+F(F(F(F(*&&F.6#F>F(F*F(F(*&&F5FAF(F0F(F(F (F2*&&%\"bGF/F(F)F(F2*&&%\"cGF/F(,(F0F(*&&F.6#F1F(F6F(F(*&&F5FMF()F',& F+F(\"\"$F2F(F(F(F2" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 41 "We subst itute the already known formulas:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "RE:=subs(\{rule2,subs(n=n+1,rule2),subs(n =n-1,rule2),rule3,subs(n=n+1,rule3),subs(n=n-1,rule3)\},RE);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#REG,**&%\"xG\"\"\",()F'%\"nGF(**F+F(,(%\" eGF(*&%\"bGF(F+F(F(F0!\"\"F(,(*&\"\"#F(%\"aGF(F1%\"dGF(*(F4F(F5F(F+F(F (F1)F',&F+F(F(F1F(F(*.F4F1F+F(,>*(\"\"$F(F0F(F.F(F(*(\"\"&F()F0F4F(F+F (F(*&F4F(F@F(F1*&)F.F4F(F+F(F(**F4F(F.F()F+F4F(F0F(F(**F?F(F.F(F+F(F0F (F1*$FCF(F1**\"\"%F(%\"cGF(F+F(F5F(F1*(FJF(F+F(F6F(F(**F4F(FJF(FEF(F5F (F(*(F4F(FJF(F5F(F(*&FJF(F6F(F1*&F@F()F+F=F(F(*(FIF(F@F(FEF(F1F(F2F1,( *&F=F(F5F(F1F6F(*(F4F(F5F(F+F(F(F1)F',&F+F(F4F1F(F(F(F(*&&F56#F+F(,()F ',&F+F(F(F(F(**FfnF(,(F.F(*&F0F(FfnF(F(F0F1F(,(*&F4F(F5F(F1F6F(*(F4F(F 5F(FfnF(F(F1F*F(F(*.F4F1FfnF(,>*(F=F(F0F(F.F(F(*(F?F(F@F(FfnF(F(*&F4F( F@F(F1*&FCF(FfnF(F(**F4F(F.F()FfnF4F(F0F(F(**F?F(F.F(FfnF(F0F(F1FGF1** FIF(FJF(FfnF(F5F(F1*(FJF(FfnF(F6F(F(**F4F(FJF(FdoF(F5F(F(*(F4F(FJF(F5F (F(FNF1*&F@F()FfnF=F(F(*(FIF(F@F(FdoF(F1F(FjnF1,(*&F=F(F5F(F1F6F(*(F4F (F5F(FfnF(F(F1F8F(F(F(F1*&&F0FYF(F)F(F1*&&FJFYF(,(F8F(**F9F(,(F.F(*&F0 F(F9F(F(F0F1F(,(*&F4F(F5F(F1F6F(*(F4F(F5F(F9F(F(F1FUF(F(*.F4F1F9F(,>*( F=F(F0F(F.F(F(*(F?F(F@F(F9F(F(*&F4F(F@F(F1*&FCF(F9F(F(**F4F(F.F()F9F4F (F0F(F(**F?F(F.F(F9F(F0F(F1FGF1**FIF(FJF(F9F(F5F(F1*(FJF(F9F(F6F(F(**F 4F(FJF(FbqF(F5F(F(*(F4F(FJF(F5F(F(FNF1*&F@F()F9F=F(F(*(FIF(F@F(FbqF(F1 F(FhpF1,(*&F=F(F5F(F1F6F(*(F4F(F5F(F9F(F(F1)F',&F+F(F=F1F(F(F(F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "re:=simplify(numer(normal(RE ))/x^(n-3)):" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 60 "Equating the hi ghest coefficient gives for monic polynomials" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "rule4:=a[n]=solve(coeff(re,x,4),a[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule4G/&%\"aG6#%\"nG\"\"\"" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 50 "and equating the second highest coeffic ient yields" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "rule5:=b[n]=factor(solve(subs(rule4,coeff(re,x,3)),b[n]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule5G/&%\"bG6#%\"nG*(,,**\"\"#\"\" \"F'F.)F)F-F.%\"aGF.!\"\"**F-F.F'F.F)F.F0F.F.*(F-F.%\"eGF.F0F.F.**F-F. F'F.F)F.%\"dGF.F1*&F4F.F6F.F1F.,&F6F.*(F-F.F0F.F)F.F.F1,(*&F-F.F0F.F1F 6F.*(F-F.F0F.F)F.F.F1" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 53 "Finall y equating the third highest coefficient yields" }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "rule6:=c[n]=factor(solve(su bs(rule5,subs(rule4,coeff(re,x,2))),c[n]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&rule6G/&%\"cG6#%\"nG,$*.F)\"\"\",(*&%\"aGF,F)F,F,%\" dGF,*&\"\"#F,F/F,!\"\"F,,<**\"\"%F,)F/F2F,)F)F2F,F'F,F,**\"\")F,F7F,F' F,F)F,F3*(F6F,F7F,F'F,F,*(F/F,)%\"bGF2F,F8F,F3*,F6F,F/F,F'F,F)F,F0F,F, **F2F,F/F,F=F,F)F,F,*&F/F,)%\"eGF2F,F,*&F/F,F=F,F3**F6F,F/F,F'F,F0F,F3 *(F=F,F0F,F)F,F3*(F>F,FCF,F0F,F3*&F'F,)F0F2F,F,*&F=F,F0F,F,F,,(F0F,F/F 3*(F2F,F/F,F)F,F,F3,(*&\"\"$F,F/F,F3F0F,*(F2F,F/F,F)F,F,F3,(*&F2F,F/F, F3F0F,*(F2F,F/F,F)F,F,!\"#F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Zeilberger's Algorithm " }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 44 "We load the package \"hsum.m pl\" from my book " }}{PARA 257 "" 0 "" {TEXT -1 64 "\"Hypergeometric \+ Summation\", Vieweg, Braunschweig/Wiesbaden, 1998" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "read \"hsum9.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\",~Maple~V~-~Maple ~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~1998-2004,~Wolfram~ Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "W e define the hypergeometric summand of the Laguerre polynomials." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "laguerr eterm:=(-1)^k/k!*binomial(n+alpha,n-k)*x^k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-laguerretermG**)!\"\"%\"kG\"\"\"-%*factorialG6#F(F'- %)binomialG6$,&%\"nGF)%&alphaGF),&F1F)F(F'F))%\"xGF(F)" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 111 "and use Zeilberger's algorithm to dete ct a recurrence equation for the sum, hence for the Laguerre polynomia ls." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "LaguerreRE:=sumrecur sion(laguerreterm,k,L(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+Lague rreREG/,(*&,(%\"nG\"\"\"%&alphaGF*F*F*F*-%\"LG6#F)F*F**&,*%\"xGF**&\" \"#F*F)F*!\"\"F+F4\"\"$F4F*-F-6#,&F)F*F*F*F*F**&,&F)F*F3F*F*-F-6#F:F*F *\"\"!" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 111 "Next, we detect the \+ differential equation of the Laguerre polynomials from their hypergeom etric representation." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "LaguerreDE:=sumdiffeq(laguerreterm,k,L(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+LaguerreDEG/,(*&%\"xG\"\"\"-%%diffG6$-%\" LG6#F(-%\"$G6$F(\"\"#F)F)*&,(F(F)%&alphaG!\"\"F)F7F)-F+6$F-F(F)F7*&F-F )%\"nGF)F)\"\"!" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 40 "Similarly, a recurrence equation w.r.t. " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" } {TEXT -1 12 " is obtained" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "sumrecursion(laguerreterm,k,L(alpha));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,(%\"nG\"\"\"%&alphaGF(F(F(F(-%\"LG6#F) F(F(*&,(%\"xGF(F)F(F(F(F(-F+6#,&F)F(F(F(F(!\"\"*&F/F(-F+6#,&F)F(\"\"#F (F(F(\"\"!" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 96 "The following com putes the recurrence equation valid for the square of the Laguerre pol ynomials " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "`rec*rec`(LaguerreRE,LaguerreRE,L(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<&,**&,T\"#5\"\"\"*&\"\"#F(%\"xGF(!\"\"*&\"#HF(%\"nGF(F (*&\"#IF()F/F*F(F(*&\"#DF()%&alphaGF*F(F(*(\"\"&F(F+F(F6F(F,*&F*F()F/ \"\"%F(F(*&\"#8F()F/\"\"$F(F(*&\"#FF(F6F(F(*&\"\"*F()F6F?F(F(*$)F6F;F( F(*&F+F(F>F(F,*(F;F(F+F(F2F(F,*(\"\"(F(F>F(F6F(F(*(FCF(F2F(F5F(F(*(\"# NF(F2F(F6F(F(*(\"#JF(F/F(F5F(F(*(F8F(F/F(FDF(F(*&F+F(FDF(F,*(F;F(F+F(F 5F(F,**F?F(F+F(F2F(F6F(F,**F?F(F+F(F/F(F5F(F,**\"\")F(F+F(F/F(F6F(F,*( F8F(F+F(F/F(F,*(\"#bF(F/F(F6F(F(F(-%\"LG6#F/F(F(*&,`o\"#mF,*&\"#qF(F+F (F(*&\"$\\\"F(F/F(F,*&\"$C\"F(F2F(F,*&\"#ZF(F5F(F,*(\"#vF(F+F(F6F(F(*& \"\"'F(F:F(F,*&\"#XF(F>F(F,*&\"#AF()F+F*F(F,*&F*F()F+F?F(F(*&\"#\"*F(F 6F(F,*&\"#6F(FDF(F,FEF,*&F\\pF(F/F(F(*(FeoF(FjoF(F2F(F,*(\"#BF(FjoF(F/ F(F,*(F`pF(F+F(F>F(F(*(\"#iF(F+F(F2F(F(*(\"#:F(F>F(F6F(F,*(\"#9F(F2F(F 5F(F,*(\"#%)F(F2F(F6F(F,*(\"#_F(F/F(F5F(F,*(FeoF(F/F(FDF(F,*&F\\pF(F6F (F(*(F?F(FjoF(F5F(F,*(\"##F(F/F(F(*&\"$g\"F(F2F(F (*&FioF(F5F(F(*(\"#[F(F+F(F6F(F,*&FeoF(F:F(F(*&\"#^F(F>F(F(*&FgqF(FjoF (F(*&F*F(F\\pF(F,*&F\\rF(F6F(F(*&F*F(FDF(F(FapF,*(FeoF(FjoF(F2F(F(*(F4 F(FjoF(F/F(F(*(F`pF(F+F(F>F(F,*(F[oF(F+F(F2F(F,*(FCF(F>F(F6F(F(*(F8F(F 2F(F5F(F(*(\"#dF(F2F(F6F(F(*(\"#@F(F/F(F5F(F(*&F/F(FDF(F(*(FeoF(FjoF(F 6F(F(*(FeoF(F+F(F5F(F,**F?F(FjoF(F/F(F6F(F(**F`pF(F+F(F2F(F6F(F,**F?F( F+F(F/F(F5F(F,**\"#YF(F+F(F/F(F6F(F,*(\"$Z\"F(F+F(F/F(F,*(\"$>\"F(F/F( F6F(F(F(-Fen6#,&F/F(F*F(F(F(*&,F(F>F(F,*&FinF(F2F(F,*&\"#**F(F/F(F,*&F >F(F6F(F,*(FVF(F2F(F6F(F,*(F`tF(F/F(F6F(F,*&FeuF(F6F(F,\"#aF,F(-Fen6#, &F/F(F?F(F(F(/-Fen6#F*,H*&#F(F;F(*&&%#_CG6#\"\"!F(&F\\wFfvF(F(F(*&#FCF ;F(*&&F\\w6#F(F(&F\\w6#F?F(F(F(*&#F?F;F(*&F[wF(FewF(F(F,*&#F?F;F(*&Fcw F(F_wF(F(F,*&FivF(**FcwF(F+F(F_wF(F6F(F(F(*&#F(F*F(**FcwF(F6F(FewF(F+F (F(F,*&FivF(**F[wF(F6F(FewF(F+F(F(F(*&FivF(*(F[wF(F5F(F_wF(F(F(*&#F(F; F(*(F[wF(F5F(FewF(F(F,*&#F(F;F(*(FcwF(F5F(F_wF(F(F,*&FivF(*(FcwF(F5F(F ewF(F(F(*&FivF(*(FcwF(FjoF(FewF(F(F(*&#F(F*F(*(F[wF(F6F(F_wF(F(F(*(F[w F(F6F(FewF(F,*&FivF(*(F[wF(FewF(F+F(F(F(*(FcwF(F6F(F_wF(F,*&#F?F*F(*(F cwF(F6F(FewF(F(F(*&#F?F*F(*(FcwF(FewF(F+F(F(F,*&FivF(*(FcwF(F+F(F_wF(F (F(/-FenF]wFjv/-FenFdwFbw" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 86 "th e following is the differential equation for the square of the Laguer re polynomials" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "`diffeq*diffeq`(LaguerreDE,LaguerreDE,L(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**&,(*(\"\"%\"\"\"%\"xGF(%\"nGF(!\"\" *(F'F(F*F(%&alphaGF(F(*&\"\"#F(F*F(F(F(-%\"LG6#F)F(F(*&,0*(F'F(F)F(F*F (F(*&F/F()F)F/F(F(*&\"\"$F(F-F(F(F(F(*&F/F()F-F/F(F(*(F'F(F)F(F-F(F+*& F'F(F)F(F+F(-%%diffG6$F0F)F(F(*&,(*&F9F(F7F(F+*(F9F(F)F(F-F(F(*&F9F(F) F(F(F(-F?6$F0-%\"$G6$F)F/F(F(*&-F?6$F0-FI6$F)F9F(F7F(F(" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 115 "and this is finally the differential e quation for the product L(n,alpha,x)*L(m,beta,x) of the Laguerre polyn omials." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`diffeq*diffeq`(LaguerreDE,subs(alpha=beta,n=m,LaguerreDE),L(x)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*&,\\t**\"\"&\"\"\")%%betaG\"\"# F()%\"nGF+F(%\"xGF(F(*,\"\"%F(F.F(F,F(%&alphaGF(F*F(F(**F+F(F*F(F,F(F. F(F(**F+F()%\"mGF+F(F1F(F.F(!\"\"*(F5F(F*F()F1F+F(F6**F+F(F4F(F*F(F.F( F6*(F5F(F8F(F)F(F(**F+F(F5F(F1F(F)F(F(*(F1F(F-F(F)F(F(**F+F(F*F(F-F(F8 F(F6**\"#7F(F5F()F.F+F(F,F(F6**F'F(F4F(F8F(F.F(F6*(F4F(F)F(F.F(F(**F?F (F4F(F@F(F-F(F(*,F?F(F5F(F@F(F-F(F1F(F6*,\"\"'F(F5F(F.F(F-F(F8F(F(*,F? F(F5F(F*F(F@F(F-F(F(*,FFF(F5F(F)F(F.F(F-F(F6*,F0F(F4F(F.F(F1F(F*F(F6*( )F1\"\"$F(F*F(F-F(F6*(FKF(F5F(F*F(F6**F0F()F.FLF(F-F(F1F(F(**F0F(FOF(F -F(F*F(F6**F0F(FOF(F5F(F1F(F(**F0F(FOF(F5F(F*F(F6**F0F(F@F(F8F(F-F(F6* (F.F(FKF(F-F(F(**F+F(F@F(F1F(F,F(F6**F0F(F@F(F5F(F)F(F(*,F0F(F@F(F1F(F -F(F*F(F6**F.F(F1F(F-F(F)F(F6*,F0F(F@F(F5F(F*F(F1F(F(**F.F(F5F(F*F(F8F (F(*,F'F(F.F(F*F(F-F(F8F(F(*,F'F(F.F(F5F(F1F(F)F(F6*(\"\")F(FOF(F,F(F( *(FhnF(FOF(F4F(F6*(F.F(F5F()F*FLF(F6**F+F(F@F(F4F(F*F(F(**FhnF(F@F(F)F (F-F(F(**F'F(F.F(F[oF(F-F(F6**\"#9F(F@F(F*F(F,F(F6**FhnF(F@F(F5F(F8F(F 6**F'F(F.F(F5F(FKF(F(**F`oF(F@F(F4F(F1F(F(*(F?F(F4F(F@F(F(*(F?F(F@F(F, F(F6*&F-F(F8F(F6*&F-F(FKF(F6**FFF(F@F(F-F(F1F(F6**F+F(F.F(F,F(F1F(F(** FFF(F.F(F-F(F8F(F(**FFF(F@F(F1F(F5F(F6**FFF(F*F(F@F(F5F(F(**FFF(F*F(F@ F(F-F(F(**FFF(F8F(F5F(F.F(F(**FFF(F)F(F5F(F.F(F6**FFF(F)F(F.F(F-F(F6*( F[oF(F1F(F-F(F(*(F[oF(F1F(F5F(F(*(F8F(F-F(F)F(F6*(F8F(F,F(F.F(F6*&)F*F 0F(F-F(F(*(F0F(F@F()F-FLF(F(*&F-F(F)F(F(*&F5F(F8F(F6*&F5F(F)F(F(*(F+F( F[oF(F-F(F(*(F0F()F5FLF(F@F(F6*&F5F()F1F0F(F6*(F+F(F5F(FKF(F6*&F5F(F[o F(F(F(-%\"LG6#F.F(F(*&,du*$F8F(F6*$F)F(F(**F+F(F5F(F.F(F*F(F6**F+F(F.F (F-F(F*F(F(**\"\"*F(F*F(F@F(F8F(F6**FFF(F[oF(F.F(F1F(F6**F+F(F.F(F5F(F 1F(F6**F0F(FOF(F-F(F1F(F(**\"#?F(FOF(F-F(F*F(F(**FbrF(FOF(F5F(F1F(F6** F0F(FOF(F5F(F*F(F6**F0F(F@F(F8F(F-F(F(**F+F(F.F(FKF(F-F(F6**FhnF(F@F(F 1F(F,F(F(**F0F(F@F(F5F(F)F(F6*,FFF(F*F(F1F(F5F(F.F(F6*,\"#;F(F@F(F1F(F -F(F*F(F6*,FFF(F.F(F1F(F-F(F)F(F(*,F[sF(F@F(F5F(F*F(F1F(F(*,FFF(F.F(F5 F(F*F(F8F(F6*,F+F(F.F(F*F(F-F(F8F(F(*,F+F(F.F(F5F(F1F(F)F(F6*(FhnF(FOF (F8F(F(*(FLF(F*F(FKF(F6*,FFF(F*F(F1F(F.F(F-F(F(*(FLF(F[oF(F1F(F(*(F[sF (FOF(F,F(F6*(F[sF(FOF(F4F(F(*(\"#KF(F5F(FOF(F6*(FhnF(F5F()F.F0F(F(**F+ F(F.F(F5F(F[oF(F(**FhnF(F@F(F4F(F*F(F6**F?F(F@F(F)F(F-F(F6**F+F(F.F(F[ oF(F-F(F(**FhnF(F@F(F*F(F,F(F(**F?F(F@F(F5F(F8F(F(**F+F(F.F(F5F(FKF(F6 **FhnF(F@F(F4F(F1F(F6*(\"#@F(F)F(F@F(F(*(F]rF(F[oF(F.F(F6*$FfpF(F(*(F? F(F4F(F@F(F6*&F+F(FKF(F6*$F`qF(F6**F]rF(F@F(F1F(F)F(F(**FFF(F.F(FKF(F* F(F(*(FhsF(FOF(F-F(F(*(F?F(F@F(F,F(F(*(FbrF(F@F(F-F(F6*(F[sF(FOF(F1F(F (*(FdtF(F@F(F8F(F6*(\"#5F(F@F(F1F(F6*(F]rF(F.F(FKF(F(*(FbuF(F.F(F8F(F( **F[sF(F@F(F-F(F1F(F6**F+F(F.F(F-F(F8F(F6**F+F(F.F(F-F(F1F(F(*(FbuF(F* F(F@F(F(*(FbuF(F)F(F.F(F6*(F+F(F8F(F*F(F6*(FbrF(F5F(F@F(F(*&F+F(F[oF(F (*(F'F(F@F(FKF(F6*(FhnF(F)F(FOF(F6*(F'F(F[oF(F@F(F(**F]rF(F.F(F1F(F)F( F6**\"#GF(F@F(F1F(F5F(F(**F]rF(F*F(F.F(F8F(F(**F[sF(F*F(F@F(F5F(F(**Fb vF(F*F(F@F(F-F(F6**FhnF(F8F(F5F(F.F(F6**F+F(F)F(F5F(F.F(F(**FhnF(F)F(F .F(F-F(F(*&F`qF(F*F(F6*&FKF(F)F(F6*(FhnF(FjsF(F-F(F6*(F0F(FjsF(F1F(F6* (F0F(F*F(FjsF(F(*&FfpF(F.F(F6*&F[oF(F8F(F(*&FfpF(F1F(F(*(F[sF(F*F(FOF( F6*(F+F(F1F(F)F(F(*&F`qF(F.F(F(F(-%%diffG6$FcqF.F(F(*&,hp**F]rF(F*F(F@ F(F8F(F(**FLF(F[oF(F.F(F1F(F(**F[sF(FOF(F-F(F1F(F6**\"#CF(FOF(F-F(F*F( F6**F]xF(FOF(F5F(F1F(F(**F[sF(FOF(F5F(F*F(F(**F+F(F@F(F8F(F-F(F(**F+F( F@F(F5F(F)F(F6*,F?F(F@F(F1F(F-F(F*F(F(*,F?F(F@F(F5F(F*F(F1F(F6*(\"#:F( FOF(F8F(F6*(FhnF(FOF(F,F(F(*(FhnF(FOF(F4F(F6*(\"#_F(F5F(FOF(F(*(FbrF(F 5F(FjsF(F6**FFF(F@F(F)F(F-F(F(**FFF(F@F(F5F(F8F(F6*(\"#DF(F)F(F@F(F6*( FFF(F[oF(F.F(F(**F]rF(F@F(F1F(F)F(F6**FLF(F.F(FKF(F*F(F6*(FixF(FOF(F-F (F6*(F[sF(F@F(F-F(F(*(\"#EF(FOF(F1F(F6*(F^yF(F@F(F8F(F(*(FhnF(F@F(F1F( F(*(FFF(F.F(FKF(F6*(\"\"(F(F.F(F8F(F6**\"#=F(F@F(F-F(F1F(F(*(FhnF(F*F( F@F(F6*(FjyF(F)F(F.F(F(*(F[sF(F5F(F@F(F6*(FjyF(F@F(FKF(F(*(FexF(F)F(FO F(F(*(FjyF(F[oF(F@F(F6**FFF(F.F(F1F(F)F(F(**F\\zF(F@F(F1F(F5F(F6**FFF( F*F(F.F(F8F(F6**F\\zF(F*F(F@F(F5F(F6**F\\zF(F*F(F@F(F-F(F(*(FbrF(FjsF( F-F(F(*(FbuF(FjsF(F1F(F(*(FbuF(F*F(FjsF(F6F^wF(*(FeyF(F*F(FOF(F(FcwF6F (-Few6$Fcq-%\"$G6$F.F+F(F(*&,J*(FhnF(F)F(FOF(F6*(FFF(F)F(F@F(F(*(FhnF( FOF(F8F(F(*(FbrF(FOF(F-F(F(*(F[sF(F5F(FjsF(F(*(F[sF(FjsF(F-F(F6*(FbrF( F5F(FOF(F6*(FbuF(FOF(F1F(F(*(FbuF(F*F(FOF(F6*(FFF(F@F(F8F(F6*(FhnF(Fjs F(F1F(F6*(FhnF(F*F(FjsF(F(**F+F(F@F(F1F(F)F(F(**FhnF(FOF(F5F(F1F(F6**F hnF(FOF(F-F(F1F(F(**F+F(F*F(F@F(F8F(F6**FhnF(FOF(F5F(F*F(F6**FhnF(FOF( F-F(F*F(F(*(F+F(F@F(FKF(F6*(F+F(F[oF(F@F(F(F(-Few6$Fcq-F_[l6$F.FLF(F(* &,.*(F+F(FjsF(F1F(F(*(F+F(F*F(FjsF(F6*&FOF(F8F(F6*&F)F(FOF(F(*(F0F(F5F (FjsF(F6*(F0F(FjsF(F-F(F(F(-Few6$Fcq-F_[l6$F.F0F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 29 "Petkovsek-van Hoeij Algorithm" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 " We would like to find a simple representation of for" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "s:=Sum(binomial(n-2*k,k)*(-4/27)^k,k=0..f loor(n/3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG-%$SumG6$*&-%)bin omialG6$,&%\"nG\"\"\"*&\"\"#F.%\"kGF.!\"\"F1F.)#!\"%\"#FF1F./F1;\"\"!- %&floorG6#,$*&\"\"$F2F-F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "summand:=binomial(n-2*k,k)*(-4/27)^k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(summandG*&-%)binomialG6$,&%\"nG\"\"\"*&\"\"#F+%\"kGF +!\"\"F.F+)#!\"%\"#FF.F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "RE:=sumrecursion(summand,k,S(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#REG/,(*(\"\"#\"\"\",&%\"nGF)\"\"$F)F)-%\"SG6#F+F)F)*(F,F),&F+F)\"\" %F)F)-F.6#,&F+F)F)F)F)F)*(\"\"*F),&F+F)F(F)F)-F.6#F8F)!\"\"\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "res:=`LREtools/hsols`(RE,S(n ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG7$)#!\"\"\"\"$%\"nG*&)# \"\"#F)F*\"\"\",&#\"\"%F)F/F*F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "result:=alpha*op(1,res)+beta*op(2,res);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'resultG,&*&%&alphaG\"\"\")#!\"\"\"\"$%\"nGF(F (*(%%betaGF()#\"\"#F,F-F(,&#\"\"%F,F(F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "sol:=solve(\{\neval(subs(Sum=sum,s=result),n=0), \neval(subs(Sum=sum,s=result),n=1)\},\n\{alpha,beta\});" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<$/%%betaG #\"\"#\"\"$/%&alphaG#\"\"\"\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "result:=subs(sol,result);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'resultG,&*&\"\"*!\"\")#F(\"\"$%\"nG\"\"\"F-**\"\"#F- F+F()#F/F+F,F-,&#\"\"%F+F-F,F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "[seq(add(binomial(n-2*k,k)*(-4/27)^k,k=0..floor(n/3)) ,n=0..10)]; n:='n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"\"F$F$#\" #B\"#F#\"#>F'#\"\"&\"\"*#\"$8$\"$H(#\"#z\"$V##\"#fF2#\"%FN\"&$o>#\"%zD F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "[seq(result,n=0..10)] ; n:='n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-\"\"\"F$F$#\"#B\"#F#\"# >F'#\"\"&\"\"*#\"$8$\"$H(#\"#z\"$V##\"#fF2#\"%FN\"&$o>#\"%zDF7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 37 "Factorization of Recurrence Equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "tau:='tau':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "read \"FactorOrder4-discrete\";" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, the name delta has been redefined\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+_Env_LRE_xG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-_Env_LRE_tauG%$tauG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "RE:=f(x+2)-(x+1)*f(x+1)+x^2*f(x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#REG,(-%\"fG6#,&%\"xG\"\"\"\"\"#F+F+*&,&F*F+F+F+F+- F'6#F.F+!\"\"*&)F*F,F+-F'6#F*F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "RE:=collect((x^2+x-1)*RE+x^3*subs(x=x+1,RE)+x*(x+1)*s ubs(x=x+2,RE),f,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG,,*( ,&%\"xG\"\"\"F)F)F),(*&\"\"$F))F(\"\"#F)F)*&\"\"'F)F(F)F)F)!\"\"F)-%\" fG6#,&F(F)F.F)F)F)*(F'F),,*$)F(\"\"%F)F)*$)F(F,F)F)*$F-F)F1F(F1F)F)F)- F36#F'F)F)*(,(F=F)F(F)F)F1F)F-F)-F36#F(F)F)*(F(F),&*&F:F)F(F)F)F,F)F)- F36#,&F(F)F,F)F)F1*(F(F)F'F)-F36#,&F(F)F:F)F)F)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "RE:=subs(\{seq(f(x+k)=tau^k,k=0..4)\},RE);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG,,*(,&%\"xG\"\"\"F)F)F),(*&\"\"$ F))F(\"\"#F)F)*&\"\"'F)F(F)F)F)!\"\"F))%$tauGF.F)F)*(F'F),,*$)F(\"\"%F )F)*$)F(F,F)F)*$F-F)F1F(F1F)F)F)F3F)F)*&,(F;F)F(F)F)F1F)F-F)F)*(F(F),& *&F8F)F(F)F)F,F)F))F3F,F)F1*(F(F)F'F))F3F8F)F)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "fact:=FactorOrder4(RE);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%factG<#,(*$)%$tauG\"\"#\"\"\"F+*&,&%\"xG!\"\"F+F/F+F )F+F+*$)F.F*F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 55 "Orthogonal Polynomial Solutions o f Recurrence Equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "re ad \"retode.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DPackage~\"REto DE\",~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyrigh t~2000-2002,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 256 18 "Example recurrence" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "RE:=P(n+2)-(x-n-1)*P(n+1)+alpha*(n+1)^2*P(n)=0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,(-%\"PG6#,&%\"nG\"\"\"\"\"#F,F ,*&,(%\"xGF,F+!\"\"F,F1F,-F(6#,&F+F,F,F,F,F1*(%&alphaGF,)F4F-F,-F(6#F+ F,F,\"\"!" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 30 "Classical continuo us solutions" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "REtoDE(RE,P(n),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%DWarnin g:~parameters~have~the~valuesG<(/%\"bG,$*&\"\"#\"\"\"%\"cGF*F*/%\"aG\" \"!/%&alphaG#F*\"\"%/F+F+/%\"dG,$*&F2F*F+F*!\"\"/%\"eGF." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%/,(*&#\"\"\"\"\"#F(*&,&*&F)F(%\"xGF(F(F(F(F(-% %diffG6$-%\"PG6$%\"nGF--%\"$G6$F-F)F(F(F(*(F)F(F-F(-F/6$F1F-F(!\"\"*(F )F(F4F(F1F(F(\"\"!7$/^#F(7$#F;F)%)infinityG/-%$rhoG6#F-,$*&F)F(-%$expG 6#,$*&F)F(F-F(F;F(F(/*&&%\"kG6#,&%\"nGF(F(F(F(&FR6#FUF;F(" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 28 "Classical discrete solutions" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "REtodis creteDE(RE,P(n),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%DWarning:~para meters~have~the~valuesG<*/%&alphaG,$*(\"\"%!\"\",&*$)%\"fG\"\"#\"\"\"F 0F0F*F0F.!\"#F0/F.F./%\"dGF4/%\"cG,**&#F0F)F0*&F-F0F4F0F0F**&#F0F)F0F4 F0F0*&#F0F/F0*(%\"gGF0F4F0F.F0F0F0*&F>F0*&F@F0F4F0F0F0/%\"aG\"\"!/F@F@ /%\"eG,$FBF*/%\"bG,&*&#F0F/F0*&F.F0F4F0F0F**&#F0F/F0F4F0F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&/,(*&#\"\"\"\"\"#F(*(,(%\"fGF(*(F)F(F,F(% \"xGF(F(F(!\"\"F(-%&DeltaG6$-%&NablaG6$-%\"PG6$%\"nG,&*&F,F(F.F(F(%\"g GF(F.F.F(F,F/F(F(**F)F(F.F(-F1F5F(,&F,F(F(F(F/F/*,F)F(F9F(F6F(F?F/F,F/ F(\"\"!7$/-%&sigmaG6#F.,**&F)F/F,F(F(F.F(#F(F)F/F " 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 }