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{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 264 15 "Maple Worksheet" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }{TEXT -1 0 "" }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 37 "Computing the Recurrence Coefficients" }}
{EXCHG {PARA 257 "" 0 "" {TEXT -1 72 "We consider the three highest co
efficients 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+*&&%'kprimeGF)F+)F-,&F*F+F+!\"\"F+F+*&
&%,kprimeprimeGF)F+)F-,&F*F+\"\"#F3F+F+" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 258 25 "We define the polynomials" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "sigma;" "6#%&sigmaG" }{TEXT -1 1 " " }{TEXT 259 3 "and" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "tau;" "6#%$tauG" }{TEXT -1 1 " " }{TEXT 260 38 "
with arbitrary coefficients a,b,c,d,e:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sigma:=a*x^2+b*x+c;\ntau:=d*x+e;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG,(*&%\"aG\"\"\")%\"xG\"\"#F(
F(*&%\"bGF(F*F(F(%\"cGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tauG,&*
&%\"dG\"\"\"%\"xGF(F(%\"eGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 61 "
The polynomial satisfies the differential equation DE=0 with:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DE:=sigma*diff(p,x$2)+tau*di
ff(p,x)+lambda[n]*p;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#DEG,(*&,(*&
%\"aG\"\"\")%\"xG\"\"#F*F**&%\"bGF*F,F*F*%\"cGF*F*,.**&%\"kG6#%\"nGF*)
F,F6F*F6F-F,!\"#F***F3F*F7F*F6F*F,F8!\"\"**&%'kprimeGF5F*)F,,&F6F*F*F:
F*F?F-F,F8F***F<F*F>F*F?F*F,F8F:**&%,kprimeprimeGF5F*)F,,&F6F*F-F:F*FE
F-F,F8F***FBF*FDF*FEF*F,F8F:F*F**&,&*&%\"dGF*F,F*F*%\"eGF*F*,(**F3F*F7
F*F6F*F,F:F***F<F*F>F*F?F*F,F:F***FBF*FDF*FEF*F,F:F*F*F**&&%'lambdaGF5
F*,(*&F3F*F7F*F**&F<F*F>F*F**&FBF*FDF*F*F*F*" }}}{EXCHG {PARA 257 "" 
0 "" {TEXT -1 24 "We collect coefficients:" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "de:=collect(simplify(DE/x^(n
-4)),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#deG,0*&,**(%\"aG\"\"\"&
%\"kG6#%\"nGF*F.F*!\"\"*&&%'lambdaGF-F*F+F*F**(F)F*F+F*)F.\"\"#F*F**(%
\"dGF*F+F*F.F*F*F*)%\"xG\"\"%F*F**&,4*&F1F*&%'kprimeGF-F*F***\"\"$F*F)
F*F>F*F.F*F/*(F5F*F)F*F>F*F**(%\"bGF*F+F*F4F*F**(F)F*F>F*F4F*F**(FDF*F
+F*F.F*F/*(%\"eGF*F+F*F.F*F**(F7F*F>F*F.F*F**&F7F*F>F*F/F*)F9FAF*F**&,
<*&FHF*F>F*F/*(\"\"'F*F)F*&%,kprimeprimeGF-F*F**(%\"cGF*F+F*F4F*F**(FT
F*F+F*F.F*F/*(F5F*FDF*F>F*F**&F1F*FQF*F**(F)F*FQF*F4F*F***\"\"&F*F)F*F
QF*F.F*F/*(FHF*F>F*F.F*F**(F7F*FQF*F.F*F**(FDF*F>F*F4F*F***FAF*FDF*F>F
*F.F*F/*(F5F*F7F*FQF*F/F*)F9F5F*F**&,2**FAF*FTF*F>F*F.F*F/*(F5F*FTF*F>
F*F**(F5F*FHF*FQF*F/*(FDF*FQF*F4F*F**(FPF*FDF*FQF*F**(FTF*F>F*F4F*F***
FZF*FDF*FQF*F.F*F/*(FHF*FQF*F.F*F*F*F9F*F***FZF*FTF*FQF*F.F*F/*(FPF*FT
F*FQF*F**(FTF*FQF*F4F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 73 "Equa
ting the highest coefficient gives the already mentioned identity for
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 "
:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "rul
e1:=lambda[n]=solve(coeff(de,x,4),lambda[n]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&rule1G/&%'lambdaG6#%\"nG,$*&F)\"\"\",(%\"aG!\"\"*&F.
F,F)F,F,%\"dGF,F,F/" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 24 "This can
 be substituted:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "de:=expand(subs(rule1,de));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%#deG,X*()%\"xG\"\"#\"\"\"%\"eGF*&%'kprimeG6#%\"nGF*!
\"\"**F)F*F(F*F+F*&%,kprimeprimeGF.F*F0**F)F*)F(\"\"$F*%\"aGF*F,F*F***
\"\"'F*F'F*F7F*F2F*F***F)F*F'F*%\"bGF*F,F*F***F9F*F(F*F;F*F2F*F***F)F*
F(F*%\"cGF*F,F*F**(F9F*F>F*F2F*F**(F5F*%\"dGF*F,F*F0**F)F*F'F*FAF*F2F*
F0*,F)F*F5F*F7F*F,F*F/F*F0*,\"\"%F*F'F*F7F*F2F*F/F*F0**F5F*F;F*&%\"kGF
.F*)F/F)F*F***F5F*F;F*FGF*F/F*F0**F'F*F;F*F,F*FIF*F**,F6F*F'F*F;F*F,F*
F/F*F0**F(F*F;F*F2F*FIF*F**,\"\"&F*F(F*F;F*F2F*F/F*F0**F'F*F>F*FGF*FIF
*F***F'F*F>F*FGF*F/F*F0**F(F*F>F*F,F*FIF*F**,F6F*F(F*F>F*F,F*F/F*F0*(F
>F*F2F*FIF*F***FOF*F>F*F2F*F/F*F0**F5F*F+F*FGF*F/F*F***F'F*F+F*F,F*F/F
*F***F(F*F+F*F2F*F/F*F*" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 81 "Equa
ting the second highest coefficient gives k'[n] as rational multiple o
f 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-F
1!\"\"F-,(*&\"\"#F-%\"aGF-F2*(F5F-F6F-F)F-F-%\"dGF-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 12 "" 1 "" {XPPMATH 20 "6#>%&rule3G/&%,kp
rimeprimeG6#%\"nG,$*&#\"\"\"\"\"#F-*,&%\"kGF(F-F)F-,>*(F.F-%\"cGF-%\"a
GF-F-**\"\"%F-F4F-F5F-F)F-!\"\"*&F4F-%\"dGF-F8*$)%\"eGF.F-F8**\"\"&F-F
=F-%\"bGF-F)F-F8*(\"\"$F-F=F-F@F-F-*(F?F-)F@F.F-F)F-F-*&F.F-FDF-F8**F.
F-F4F-)F)F.F-F5F-F-*(F4F-F)F-F:F-F-*(F7F-FDF-FGF-F8*&F<F-F)F-F-**F.F-F
=F-FGF-F@F-F-*&FDF-)F)FBF-F-F-,(*&F.F-F5F-F8*(F.F-F5F-F)F-F-F:F-F8,(*(
F.F-F5F-F)F-F-*&FBF-F5F-F8F:F-F8F-F-" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT -1 60 "Without loss of generality we consider the monic case, he
nce" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "k[
n]:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"kG6#%\"nG\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 262 13 "and therefore" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 6 "rule2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&
%'kprimeG6#%\"nG*(F'\"\"\",(%\"eGF)*&%\"bGF)F'F)F)F-!\"\"F),(*&\"\"#F)
%\"aGF)F.*(F1F)F2F)F'F)F)%\"dGF)F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "rule3;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%,kprimepr
imeG6#%\"nG,$*,\"\"#!\"\"F'\"\"\",>*(F*F,%\"cGF,%\"aGF,F,**\"\"%F,F/F,
F0F,F'F,F+*&F/F,%\"dGF,F+*$)%\"eGF*F,F+**\"\"&F,F7F,%\"bGF,F'F,F+*(\"
\"$F,F7F,F:F,F,*(F9F,)F:F*F,F'F,F,*&F*F,F>F,F+**F*F,F/F,)F'F*F,F0F,F,*
(F/F,F'F,F4F,F,*(F2F,F>F,FAF,F+*&F6F,F'F,F,**F*F,F7F,FAF,F:F,F,*&F>F,)
F'F<F,F,F,,(*&F*F,F0F,F+*(F*F,F0F,F'F,F,F4F,F+,(*(F*F,F0F,F'F,F,*&F<F,
F0F,F+F4F,F+F," }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 94 "We would like
 to compute the coefficients a(n), b(n) and c(n) in the recurrence equ
ation 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#%\"nGF(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:=subs(\{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(\"\"#F2F(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(\"\"$F2F(F(F(
F2" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 41 "We substitute the already
 known formulas:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 99 "RE:=subs(\{rule2,subs(n=n+1,rule2),subs(n=n-1,rule2),
rule3,subs(n=n+1,rule3),subs(n=n-1,rule3)\},RE);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%#REG,**&%\"xG\"\"\",()F'%\"nGF(**F+F(,(%\"eGF(*&%\"bG
F(F+F(F(F0!\"\"F(,(*&\"\"#F(%\"aGF(F1*(F4F(F5F(F+F(F(%\"dGF(F1)F',&F+F
(F(F1F(F(*.F4F1F+F(,>*(F4F(%\"cGF(F5F(F(**\"\"%F(F=F(F5F(F+F(F1*&F=F(F
7F(F1*$)F.F4F(F1**\"\"&F(F.F(F0F(F+F(F1*(\"\"$F(F.F(F0F(F(*(FDF()F0F4F
(F+F(F(*&F4F(FHF(F1**F4F(F=F()F+F4F(F5F(F(*(F=F(F+F(F7F(F(*(F?F(FHF(FK
F(F1*&FBF(F+F(F(**F4F(F.F(FKF(F0F(F(*&FHF()F+FFF(F(F(F2F1,(*(F4F(F5F(F
+F(F(*&FFF(F5F(F1F7F(F1)F',&F+F(F4F1F(F(F(F(*&&F56#F+F(,()F',&F+F(F(F(
F(**FfnF(,(F.F(*&F0F(FfnF(F(F0F1F(,(*&F4F(F5F(F1*(F4F(F5F(FfnF(F(F7F(F
1F*F(F(*.F4F1FfnF(,>*(F4F(F=F(F5F(F(**F?F(F=F(F5F(FfnF(F1F@F1FAF1**FDF
(F.F(F0F(FfnF(F1*(FFF(F.F(F0F(F(*(FDF(FHF(FfnF(F(*&F4F(FHF(F1**F4F(F=F
()FfnF4F(F5F(F(*(F=F(FfnF(F7F(F(*(F?F(FHF(FfoF(F1*&FBF(FfnF(F(**F4F(F.
F(FfoF(F0F(F(*&FHF()FfnFFF(F(F(FjnF1,(*(F4F(F5F(FfnF(F(*&FFF(F5F(F1F7F
(F1F8F(F(F(F1*&&F0FYF(F)F(F1*&&F=FYF(,(F8F(**F9F(,(F.F(*&F0F(F9F(F(F0F
1F(,(*&F4F(F5F(F1*(F4F(F5F(F9F(F(F7F(F1FUF(F(*.F4F1F9F(,>*(F4F(F=F(F5F
(F(**F?F(F=F(F5F(F9F(F1F@F1FAF1**FDF(F.F(F0F(F9F(F1*(FFF(F.F(F0F(F(*(F
DF(FHF(F9F(F(*&F4F(FHF(F1**F4F(F=F()F9F4F(F5F(F(*(F=F(F9F(F7F(F(*(F?F(
FHF(FdqF(F1*&FBF(F9F(F(**F4F(F.F(FdqF(F0F(F(*&FHF()F9FFF(F(F(FhpF1,(*(
F4F(F5F(F9F(F(*&FFF(F5F(F1F7F(F1)F',&F+F(FFF1F(F(F(F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "re:=simplify(numer(normal(RE))/x^(n
-3));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#reG,`ep**\"\"#\"\"\"&%\"cG
6#%\"nGF(F*F()%\"dG\"\"&F(!\"\"**\"#7F(F)F()%\"bGF'F()F.\"\"%F(F0*,F'F
(%\"xGF(F)F(%\"eGF(F-F(F(*,\"%#>#F()F8F'F(F)F()%\"aG\"\"'F()F,F'F(F0*.
\"$)GF()F8\"\"$F()F,FDF(F4F()F.FDF()F>F'F(F0*,\"%'4\"F(FCF(FEF(F4F()F>
F/F(F(*,F'F(FCF(F@F(F4F(F-F(F(*,\"$S#F(FCF(F@F(F4F(FJF(F0*,F'F(FCF(F,F
(F4F(F-F(F0*.\"$=\"F(FCF(F,F(F4F(FGF(FFF(F0*.\"$C'F(FCF(FEF(F9F()F.F'F
()F>FDF(F0*,\"$c#F(F)F(F*F()F,F?F(FJF(F(*,FDF(F)F(F*F(F,F(F-F(F(**F'F(
F<F(F)F()F.F?F(F0*0\"%[6F(F8F(&F4F+F()F,F6F(F3F(FTF(F.F(F(*.\"%g6F(F)F
(F*F(FSF(FTF(F@F(F(*0\"%'R#F(F8F(F)F(F4F(FSF(FTF(FEF(F0*0\"%aEF(F8F(F)
F(F4F(FSF(FTF(F@F(F(*0\"$)oF(F8F(F)F(F9F(FSF(FTF(FEF(F(*0\"%?5F(F8F(F)
F(F9F(FSF(FTF(F@F(F0*.\"%+7F()F8F6F(&F>F+F(FTF(FFF(F@F(F(*.\"%!3%F(Feo
F(FfoF()F>F6F(FSF(F@F(F0*.\"%+CF(FeoF(FfoF(FioF(FSF(FEF(F(*.\"$![F(Feo
F(FfoF(FioF(FhnF(FSF(F0*.\"$?$F(FeoF(FfoF(FTF(FEF(FFF(F0*.\"%;9F(FCF(F
EF(F9F(FioF(F.F(F(*0\"$'\\F(F)F(F9F()F,F/F(F4F(FTF(F.F(F(*0\"#kF(F)F(F
9F(FWF(F4F(FTF(F.F(F0*.\"#!)F(F<F(FdpF(F*F(FSF(FTF(F(*.\"#gF(F)F(F9F(F
4F(FFF(F>F(F0*0\"$a&F(F)F(F9F(F4F(FTF(F,F(F.F(F(*0\"%A;F(F)F(F9F(F4F(F
TF(F@F(F.F(F0*0\"%)[\"F(F)F(F9F(F4F(FhnF(FTF(F.F(F0*,\"#KF(F)F(F*F()F,
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(F3F(F5F(F(*.\"$)=F(F<F(FfoF(F@F(FgqF(FioF(F(*.\"$?\"F(F<F(FfoF(F,F(Fg
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(F)F(F*F(FioF(F.F(F0*,\"#?F(F)F(F*F(F5F(F>F(F(*0\"$g\"F(F8F(F)F(F9F(Fh
nF(FSF(FTF(F0*0FirF(F8F(F)F(F4F(FdpF(FSF(FTF(F0*.F]pF(F)F(F*F(FhnF(FSF
(FTF(F(*.FhpF(F)F(F*F(FdpF(FSF(FTF(F0*.\"%[9F(F)F(F9F(F4F(FioF(FEF(F0*
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(FioF(F0*.F;F(FCF(FgnF(FJF(F.F(F,F(F0*.Ff^mF(FCF(FgnF(F5F(FGF(F,F(F(*.
FitF(FCF(FgnF(F-F(F>F(F,F(F0*.FdoF(FCF(FgnF(FFF(FTF(F@F(F(*.FhoF(FCF(F
gnF(FioF(FSF(F@F(F0*.F]pF(FCF(FgnF(FioF(FhnF(FSF(F0*.F[pF(FCF(FgnF(Fio
F(FEF(FSF(F(*.F\\rF(FCF(FgnF(FGF(F@F(F5F(F0*.Fh_lF(FCF(FgnF(FJF(F@F(F.
F(F(*.Fj]lF(FCF(FgnF(FJF(FEF(F.F(F0*.F[pF(FCF(FgnF(FJF(FhnF(F.F(F(*.F_
rF(FCF(FgnF(FJF(FdpF(F.F(F0*.Ff^mF(F<F(F)F(FGF(F5F(F,F(F(*.Fh`lF(F<F(F
hnF(F*F(FSF(FTF(F0*.FfvF(F<F(F@F(FgqF(FSF(FGF(F(*.FjuF(F<F(FEF(F3F(FTF
(F.F(F(*,FchlF(FCF(FdpF(F4F(FJF(F(*,F\\[lF(FeoF(FfoF(F=F(FdpF(F(*,FbyF
(FeoF(FfoF(FioF(FSF(F0*,F;F(FeoF(FfoF(F=F(F@F(F0*,F\\clF(FeoF(FfoF(FGF
(F5F(F0*.FhpF(F<F(FEF(F3F(FFF(F>F(F(*.\"#eF(F<F(F,F(F3F(FSF(FGF(F0**F<
F(F@F(F*F(F-F(F(*.F\\gmF(F<F(F@F(F*F(FioF(F.F(F(*.\"$q&F(F)F(F*F(FTF(F
SF(F,F(F0*0\"$9'F(F8F(F)F(F9F(FTF(FSF(F,F(F(*.\"$^)F(F)F(F3F(FhnF(FSF(
FGF(F0*.\"$G#F(F)F(F3F(FdpF(FSF(FGF(F(*.\"$o'F(F)F(F9F(F4F(FioF(F@F(F(
*.F`anF(FCF(F@F(F9F(FSF(FTF(F(*.FcxF(F<F(FWF(F3F(FTF(F.F(F0*0FgrF(F8F(
F)F(F4F(FEF(F5F(F>F(F0*0F\\hmF(F8F(F)F(F4F(F@F(F5F(F>F(F(*.\"$+#F(F)F(
F*F(FEF(FFF(FGF(F(*0\"$;\"F(F<F(FfoF(FEF(FgqF(FTF(F.F(F0*0F]wF(F8F(Fgn
F(FdpF(F*F(FioF(F.F(F(*0FhpF(F8F(FgnF(FWF(F*F(FioF(F.F(F0*0FerF(F)F(F9
F(FdpF(F4F(FSF(FGF(F0*0F_uF(F)F(F9F(FhnF(F4F(FFF(F>F(F0*.F'F(F)F(F9F(F
EF(F4F(F5F(F0*.\"$5\"F(F)F(F9F(F4F(FGF(FSF(F(*.FjpF(F)F(F9F(F4F(FTF(F.
F(F0*.\"$'pF(F)F(F3F(FTF(F,F(F.F(F0*.FitF(F)F(FgqF(FhnF(FGF(FSF(F0*.F[
`lF(F<F(F,F(FgqF(FFF(F>F(F(*2F^vF(F<F(FfoF(FEF(F9F(F4F(FTF(F.F(F0*2Ffp
F(F<F(FfoF(FWF(F9F(F4F(FTF(F.F(F0*2\"$Q%F(F<F(FfoF(F@F(F9F(F4F(FTF(F.F
(F(*.FMF(F<F(FWF(F9F(F4F(FioF(F0*0F`elF(F<F(FEF(F9F(F4F(FGF(FSF(F(*,Fg
uF(F<F(FhnF(F*F(FJF(F0*.FhpF(FCF(FhnF(F4F(FFF(FGF(F(*0F]blF(F<F(F@F(F9
F(F4F(FTF(F.F(F(*.FhpF(F8F(FgnF(F@F(F*F(FJF(F(*.F^elF(F<F(F@F(F3F(FGF(
FSF(F(*0FhglF(F8F(FgnF(F,F(FgqF(FSF(FGF(F(*0FfalF(F8F(FgnF(F,F(F3F(F.F
(FTF(F0*0FicnF(F8F(FgnF(F,F(F3F(FSF(FGF(F(*2FjpF(F8F(FgnF(F,F(F9F(F4F(
F.F(FTF(F(*2\"#()F(F8F(FgnF(F,F(F9F(F4F(FSF(FGF(F0*0F/F(F8F(FgnF(F@F(F
9F(F4F(F5F(F(*0FDF(F8F(FgnF(F,F(F9F(F4F(F5F(F0*0\"$X\"F(F8F(FgnF(F@F(F
*F(FFF(FGF(F0*0Fg`mF(F<F(FgnF(F@F(F4F(FFF(FGF(F0*0FgrF(F8F(FgnF(F,F(Fg
qF(F.F(FTF(F0*0FfalF(F8F(FgnF(F,F(F*F(FioF(F.F(F(*0FhyF(F8F(FgnF(F,F(F
*F(FTF(FSF(F0*0Fc^nF(F8F(FgnF(F@F(F*F(FTF(FSF(F(*0FexF(F8F(FgnF(F,F(F*
F(FGF(FFF(F(*0FjfnF(F<F(F,F(F9F(F4F(FSF(FGF(F(*0FjpF(F<F(FEF(F9F(F4F(F
FF(F>F(F0*.F^gnF(F<F(F@F(F*F(FFF(FGF(F(*2F\\tF(F<F(FfoF(F@F(F9F(F4F(FF
F(F>F(F(*2Fg[lF(F<F(FfoF(F,F(F9F(F4F(FTF(F.F(F0*2F^_lF(F<F(FfoF(F,F(F9
F(F4F(FFF(F>F(F0*0F\\^nF(F<F(FgnF(F@F(F4F(FioF(F.F(F0*0Fc\\mF(F<F(FgnF
(F,F(F4F(F5F(F>F(F0*0FjuF(F8F(FgnF(FEF(F*F(FioF(F.F(F(*0F\\gmF(F8F(Fgn
F(F@F(F*F(FioF(F.F(F0*.FgrF(FCF(FEF(F4F(F5F(F>F(F(*.Fi\\nF(FCF(FEF(F4F
(FioF(F.F(F0*.FbuF(F8F(FgnF(FhnF(FgqF(FioF(F0*.F_tF(F8F(FgnF(FEF(FgqF(
FioF(F(*.FielF(F8F(FgnF(FdpF(FgqF(FioF(F(*.FfalF(F8F(FgnF(F@F(FgqF(Fio
F(F0*.F\\\\mF(F8F(FgnF(FWF(F*F(FJF(F(*.Fe\\lF(FCF(FfoF(F9F(FSF(FTF(F(*
.F\\clF(FCF(FfoF(F9F(FFF(FGF(F0*.FbyF(FCF(FfoF(F9F(FioF(F.F(F0*.F`vF(F
CF(FfoF(F9F(F5F(F>F(F(*0F]dnF(FCF(FfoF(F9F(FSF(FTF(F,F(F0*0Ff^mF(FCF(F
foF(F9F(FSF(FTF(F@F(F0*0FidmF(FCF(FfoF(F9F(FEF(FSF(FTF(F(*0FirF(FCF(Ff
oF(F9F(FhnF(FSF(FTF(F0*0Ff^mF(FCF(FfoF(F4F(FEF(FSF(FTF(F0*0F]dnF(FCF(F
foF(F4F(F@F(FTF(FSF(F0*0Fj`lF(FCF(FfoF(F4F(F@F(FGF(FFF(F(*,\"#GF(F)F(F
3F(F,F(F5F(F(" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 60 "Equating the h
ighest 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 \+
coefficient 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,$*(,,
**\"\"#\"\"\"F'F/)F)F.F/%\"aGF/F/**F.F/F'F/F)F/F1F/!\"\"*(F.F/%\"eGF/F
1F/F3**F.F/F'F/F)F/%\"dGF/F/*&F5F/F7F/F/F/,(*&F.F/F1F/F3*(F.F/F1F/F)F/
F/F7F/F3,&*(F.F/F1F/F)F/F/F7F/F3F3" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 
-1 53 "Finally equating the third highest coefficient yields" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "rule6:=
c[n]=factor(solve(subs(rule5,subs(rule4,coeff(re,x,2))),c[n]));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&rule6G/&%\"cG6#%\"nG,$*.F)\"\"\",(%
\"dGF,*&\"\"#F,%\"aGF,!\"\"*&F1F,F)F,F,F,,<**\"\"%F,)F1F0F,)F)F0F,F'F,
F,**\"\")F,F7F,F'F,F)F,F2*(F6F,F7F,F'F,F,*(F1F,)%\"bGF0F,F8F,F2**F0F,F
1F,F=F,F)F,F,*,F6F,F1F,F'F,F)F,F.F,F,**F6F,F1F,F'F,F.F,F2*&F1F,)%\"eGF
0F,F,*&F1F,F=F,F2*(F=F,F.F,F)F,F2*&F=F,F.F,F,*&F'F,)F.F0F,F,*(FDF,F>F,
F.F,F2F,,(*(F0F,F1F,F)F,F,*&\"\"$F,F1F,F2F.F,F2,(*(F0F,F1F,F)F,F,F1F2F
.F,F2,(*&F0F,F1F,F2*(F0F,F1F,F)F,F,F.F,!\"#F2" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Zeilbe
rger's algorithm" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 44 "We load the \+
package \"hsum.mpl\" from my book " }}{PARA 257 "" 0 "" {TEXT -1 64 "
\"Hypergeometric Summation\", Vieweg, Braunschweig/Wiesbaden, 1998" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read \"hsum9.mpl\";" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summation\"
,~Maple~V~-~Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyright~19
98-2004,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 65 "We define the hypergeometric summand of the Laguerre p
olynomials." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 69 "laguerreterm:=pochhammer(alpha+1,n)/n!*hyperterm([-n],[alpha+1
],x,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-laguerretermG*.-%+pochha
mmerG6$,&%&alphaG\"\"\"F+F+%\"nGF+-%*factorialG6#F,!\"\"-F'6$,$F,F0%\"
kGF+-F'6$F)F4F0)%\"xGF4F+-F.6#F4F0" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 
-1 111 "and use Zeilberger's algorithm to detect a recurrence equation
 for the sum, hence for the Laguerre polynomials." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 38 "RE:=sumrecursion(laguerreterm,k,L(n));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,(*&,(%\"nG\"\"\"F*F*%&alphaGF*
F*-%\"LG6#F)F*F**&,*F+!\"\"\"\"$F1%\"xGF**&\"\"#F*F)F*F1F*-F-6#,&F)F*F
*F*F*F**&,&F5F*F)F*F*-F-6#F:F*F*\"\"!" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT -1 111 "Next, we detect the differential equation of the Laguerr
e polynomials from their hypergeometric representation." }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "DE:=sumdiffeq(lagu
erreterm,k,L(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/,(*&%\"xG
\"\"\"-%%diffG6$-%\"LG6#F(-%\"$G6$F(\"\"#F)F)*&,(F(F)%&alphaG!\"\"F)F7
F)-F+6$F-F(F)F7*&F-F)%\"nGF)F)\"\"!" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT -1 97 "To compute recurrence and differential equations for sums
 and products, we load the gfun package." }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun):" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT -1 96 "The following computes the recurrence e
quation valid for the square of the Laguerre polynomials " }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "`rec*rec`(RE,RE,
L(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<&,**&,T\"#5\"\"\"*&\"\"#F(%
\"xGF(!\"\"*&\"#HF(%\"nGF(F(*(\"\"%F(F+F()F/F*F(F,*&F+F()F/\"\"$F(F,*&
\"#DF()%&alphaGF*F(F(*$)F9F1F(F(*&\"\"*F()F9F5F(F(*&\"#8F(F4F(F(*&F*F(
)F/F1F(F(*&\"#FF(F9F(F(*(\"#bF(F/F(F9F(F(*(\"\"&F(F+F(F9F(F,*(FHF(F+F(
F/F(F,*&\"#IF(F2F(F(*(FHF(F/F(F>F(F(*(\"#JF(F/F(F8F(F(*(F=F(F2F(F8F(F(
*(\"#NF(F2F(F9F(F(*(\"\"(F(F4F(F9F(F(*&F>F(F+F(F,*(F1F(F8F(F+F(F,**F5F
(F/F(F8F(F+F(F,**F5F(F2F(F9F(F+F(F,**\"\")F(F/F(F9F(F+F(F,F(-%\"LG6#F/
F(F(*&,`o\"#mF,*&\"#AF()F+F*F(F,*&\"#qF(F+F(F(*&\"$\\\"F(F/F(F,*(\"#BF
(F/F(F\\oF(F,*(\"#iF(F+F(F2F(F(*&)F+F5F(F/F(F(*(\"\"'F(F\\oF(F2F(F,*(
\"#6F(F+F(F4F(F(*&\"#ZF(F8F(F,*&FfoF(F9F(F(*(\"#<F(F9F(F\\oF(F,*&F*F(F
foF(F(F:F,*&FjoF(F>F(F,*&\"#XF(F4F(F,*&FhoF(FBF(F,*&\"#\"*F(F9F(F,*(\"
$`\"F(F/F(F9F(F,*(\"#vF(F+F(F9F(F(*(\"$:\"F(F+F(F/F(F(*&\"$C\"F(F2F(F,
*(FhoF(F/F(F>F(F,*(\"#_F(F/F(F8F(F,*(\"#9F(F2F(F8F(F,*(\"#%)F(F2F(F9F(
F,*(\"#:F(F4F(F9F(F,*(F5F(F>F(F+F(F(*(\"#EF(F8F(F+F(F(*(F5F(F8F(F\\oF(
F,**FcqF(F/F(F8F(F+F(F(**F=F(F/F(F9F(F\\oF(F,**F[oF(F2F(F9F(F+F(F(**\"
##)F(F/F(F9F(F+F(F(F(-Fen6#,&F/F(F(F(F(F(*&,hn\"$5\"F(*&FjqF(F\\oF(F(*
&\"$-\"F(F+F(F,*&\"$>#F(F/F(F(*(F7F(F/F(F\\oF(F(*(F^oF(F+F(F2F(F,FeoF,
*(FhoF(F\\oF(F2F(F(*(FjoF(F+F(F4F(F,*&F[oF(F8F(F(*(FhoF(F9F(F\\oF(F(*&
F*F(FfoF(F,*&F*F(F>F(F(*&\"#^F(F4F(F(*&FhoF(FBF(F(*&F`rF(F9F(F(*(\"$>
\"F(F/F(F9F(F(*(\"#[F(F+F(F9F(F,*(\"$Z\"F(F+F(F/F(F,*&\"$g\"F(F2F(F(*&
F/F(F>F(F(*(\"#@F(F/F(F8F(F(*(FHF(F2F(F8F(F(*(\"#dF(F2F(F9F(F(*(F=F(F4
F(F9F(F(*(FhoF(F8F(F+F(F,**F5F(F/F(F8F(F+F(F,**F5F(F/F(F9F(F\\oF(F(**F
joF(F2F(F9F(F+F(F,**\"#YF(F/F(F9F(F+F(F,F(-Fen6#,&F*F(F/F(F(F(*&,<*(FY
F(F2F(F9F(F,*&F4F(F9F(F,*(FbtF(F/F(F9F(F,*&\"#=F(F9F(F,*&FinF(F2F(F,*&
\"#>F(F4F(F,*&\"#**F(F/F(F,\"#aF,*(FYF(F+F(F2F(F(F3F(*(FbtF(F+F(F/F(F(
*&FfuF(F+F(F(*&F*F(FBF(F,F(-Fen6#,&F/F(F5F(F(F(/-Fen6#F*,H*&#F(F1F(*(&
%#_CG6#F(F(F+F(&F\\wFfvF(F(F(*&FivF(*(&F\\w6#\"\"!F(&F\\w6#F5F(F+F(F(F
(*(FawF(F9F(FdwF(F,*&#F(F*F(*(FawF(F9F(F^wF(F(F(*&FivF(*(F[wF(F8F(FdwF
(F(F(*&#F(F1F(*(F[wF(F8F(F^wF(F(F,*&FivF(*(F[wF(F\\oF(FdwF(F(F(*&#F(F1
F(*(FawF(F8F(FdwF(F(F,*&#F(F*F(**F[wF(F+F(FdwF(F9F(F(F,*&FivF(**F[wF(F
+F(F^wF(F9F(F(F(*&FivF(**FawF(F9F(FdwF(F+F(F(F(*&#F5F*F(*(F[wF(FdwF(F9
F(F(F(*(F[wF(F^wF(F9F(F,*&#F5F*F(*(F[wF(F+F(FdwF(F(F,*&#F5F1F(*&F[wF(F
^wF(F(F,*&#F5F1F(*&FawF(FdwF(F(F,*&FivF(*(FawF(F8F(F^wF(F(F(*&FivF(*&F
awF(F^wF(F(F(*&#F=F1F(*&F[wF(FdwF(F(F(/-FenFbwF[z/-FenF]wF^z" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT -1 81 "and this is the differential equ
ation for the square of the Laguerre polynomials." }{MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "`diffeq*diffeq`(DE,DE,L(x
));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**&,(*&\"\"#\"\"\"%\"nGF(F(*(
\"\"%F(%\"xGF(F)F(!\"\"*(F+F(F)F(%&alphaGF(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(*&,(*&F7F(F9F(F-*(F7F(F,F(F/F(F(
*&F7F(F,F(F(F(-F?6$F0-%\"$G6$F,F'F(F(*&-F?6$F0-FI6$F,F7F(F9F(F(" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT -1 51 "Next, we define the summand of t
he Hahn polynomials" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 110 "hahnterm:=(-1)^n*pochhammer(N-n,n)*pochhammer(beta+1
,n)/n!*hyperterm([-n,-x,alpha+beta+n+1],[beta+1,1-N],1,k);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%)hahntermG*6)!\"\"%\"nG\"\"\"-%+pochhammer
G6$,&%\"NGF)F(F'F(F)-F+6$,&%%betaGF)F)F)F(F)-%*factorialG6#F(F'-F+6$,$
F(F'%\"kGF)-F+6$,$%\"xGF'F9F)-F+6$,*%&alphaGF)F2F)F(F)F)F)F9F)-F+6$F1F
9F'-F+6$,&F)F)F.F'F9F'-F46#F9F'" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 
42 "and compute a recurrence equation for them" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sumrecursion(hahnterm,k,h(n)
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*.,(%\"nG\"\"\"F(F(%\"NG!\"\"
F(,(F'F(F(F(%&alphaGF(F(,(F'F(F(F(%%betaGF(F(,*F,F(\"\"%F(*&\"\"#F(F'F
(F(F.F(F(,,F(F(F)F(F,F(F.F(F'F(F(-%\"hG6#F'F(F(*(,**&F2F(F'F(F(\"\"$F(
F,F(F.F(F(,^oF0F(*(\"\"'F(%\"xGF(F.F(F(*&\"\")F(F>F(F(*&F=F(F'F(F(*&F,
F(F.F(F(*&F2F()F.F2F(F(*(\"\"&F(F.F(F'F(F(**F0F(F>F(F.F(F'F(F(*(F0F(F>
F()F'F2F(F(*&FDF(F'F(F(*&F.F(FIF(F(*(F=F(F'F(F)F(F**(F2F(FIF(F)F(F**&F
DF(F)F(F***F2F(F.F(F'F(F)F(F***F2F(F'F(F,F(F)F(F**(F.F(F,F(F)F(F**$)F,
F2F(F**(F:F(F.F(F)F(F**&FFF(F.F(F(F,F(*&F'F(F,F(F**(F=F(F>F(F,F(F(*(\"
#7F(F>F(F'F(F(*&F>F(FDF(F(*(F:F(F,F(F)F(F**&F2F(FIF(F(*&F'F(FSF(F**&FI
F(F,F(F**&FSF(F>F(F(**F2F(F>F(F,F(F.F(F(**F0F(F'F(F,F(F>F(F(*&F0F(F)F(
F*F(-F56#,&F'F(F(F(F(F(**,&F2F(F'F(F(,*F,F(F.F(F'F(F2F(F(,**&F2F(F'F(F
(F,F(F.F(F2F(F(-F56#FaoF(F*\"\"!" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 
-1 53 "Similarly, a difference equation w.r.t. x is obtained" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sumrecu
rsion(hahnterm,k,h(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*(,&%\"x
G\"\"\"F(F(F(,*F'F(%&alphaG!\"\"F(F(%\"NGF+F(-%\"hG6#F'F(F(*&,>*&\"\"'
F(F'F(F(*&\"\"#F()F'F5F(F(*&F5F(%%betaGF(F(F*F+*&F'F(F*F(F+*(F5F(F'F(F
,F(F+\"\"&F(*&F8F(F,F(F+*&\"\"$F(F,F(F+%\"nGF(*&F?F(F*F(F(*&F'F(F8F(F(
*$)F?F5F(F(*&F8F(F?F(F(F(-F.6#F&F(F+*(,(F'F(F5F(F,F+F(,(F'F(F8F(F5F(F(
-F.6#,&F'F(F5F(F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 55 "Orthogonal Polynomial Solut
ions of Recurrence Equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "read \"retode.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DPackage~
\"REtoDE\",~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCo
pyright~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#,&\"\"#\"\"\"%
\"nGF,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 c
ontinuous solutions" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "REtoDE(RE,P(n),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
$%DWarning:~parameters~have~the~valuesG<(/%\"aG\"\"!/%\"bG,$*&\"\"#\"
\"\"%\"cGF-F-/%&alphaG#F-\"\"%/%\"eGF'/F.F./%\"dG,$*&F2F-F.F-!\"\"" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#7%/,(*&#\"\"\"\"\"#F(*&,&*&F)F(%\"xGF(
F(F(F(F(-%%diffG6$-%\"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(-%$expG6#,$*&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 "REtodi
screteDE(RE,P(n),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%DWarning:~par
ameters~have~the~valuesG<*/%\"fGF&/%&alphaG,$*(\"\"%!\"\",&*$)F&\"\"#
\"\"\"F1F1F,F1F&!\"#F1/%\"cG,**&#F1F+F1*&F/F1%\"dGF1F1F,*&#F1F+F1F9F1F
1*&#F1F0F1*(%\"gGF1F9F1F&F1F1F1*&F=F1*&F?F1F9F1F1F1/%\"eG,$FAF,/F?F?/%
\"aG\"\"!/%\"bG,&*&#F1F0F1*&F&F1F9F1F1F,*&#F1F0F1F9F1F,/F9F9" }}{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<F//,&FDF(-%$tauGFFF(
,$**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(/-%$rhoG
6#%\"xG)*&FPF(F?F/F./*&&%\"kG6#,&%\"nGF(F(F(F(&Fhn6#F[oF/*&F(F(F,F/" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "strict:=true;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%'strictG%%trueG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 263 59 "Without translations no classical discrete solutions exi
st." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "REtodiscreteDE(RE,P(
n),x);" }}{PARA 8 "" 1 "" {TEXT -1 110 "Error, (in REtodiscreteDE) thi
s recurrence equation has no classical discrete orthogonal polynomial \+
solutions\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "3 10 1 0" 80 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }