{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{PSTYLE " Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{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 -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 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Map le Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 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 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Maple package retode, see \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Koepf , W., Schmersau, D.: (1998)." }}{PARA 0 "" 0 "" {TEXT -1 73 "Recurrenc e Equations and their Classical Orthogonal Polynomial Solutions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "To be obt ained from koepf@imn.htwk-leipzig.de" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "read retode;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "classical contin uous orthogonal polynomial solutions" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "do not search for linear transforms or pa rameters" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "strict:=true:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Example 2.1" }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "RE:=(n+3)*p(n+2)-x*(n+2)*p( n+1)+(n+1)*p(n)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,(*&,&%\" nG\"\"\"\"\"$F*F*-%\"pG6#,&F)F*\"\"#F*F*F**(%\"xGF*F/F*-F-6#,&F)F*F*F* F*!\"\"*&F5F*-F-6#F)F*F*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "REtoDE(RE,p(n),x);" }}{PARA 6 "" 1 "" {TEXT -1 32 "Warning: seve ral solutions found" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7'7$/,(*(,&%\"x G\"\"\"!\"#F*F*,&F)F*\"\"#F*F*-%%diffG6$-%\"pG6$%\"nGF)-%\"$G6$F)F-F*F **&,&F)F*!\"\"F*F*-F/6$F1F)F*F-*(F4F*,&F4F*F*F*F*F1F*F:\"\"!7$/%\"IG7$ F+F-/-%$rhoG6#F)*&*$-%%sqrtG6#F,\"\"\"FM*$-FK6#F(FM!\"\"7$/,(F'F**&,&F )F*F*F*F*F;FMF-F=F:F?7$FA/FE*&*$-FK6#F(FMFM*$-FK6#F,FMFQ7$/,(F'F**&F)F *F;FMF**&)F4F-FMF1FMF:F?7$FA/FE*&*$-FK6#,&*$)F)F-FMF*!\"%F*FMFM*&F(\" \"\"F,\"\"\"FQ7$/,(F'F*F]o\"\"$*(F4FM,&F4F*F-F*F*F1FMF:F?7$FA/FE*$FdoF M/*&&%\"kG6#,&F4F*F*F*FM&Fip6#F4FQ*&F>FMFbpFQ" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 282 "This result is interpreted as follows: The recurrence equation RE has four classical orthogonal polynomial solutions which \+ satisfy the above four differential equations. All four resulting poly nomial systems are supported in the interval I=[-2,2], having the give n weight functions " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 61 "(x ). The term ratio for the leading coefficients is given as " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "k[n+1]/k[n]=(n+1)/(n+2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"kG6#,&%\"nG\"\"\"F*F*\"\"\"&F&6 #F)!\"\"*&F(F+,&F)F*\"\"#F*F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "h ence " }{XPPEDIT 18 0 "k[n];" "6#&%\"kG6#%\"nG" }{TEXT -1 12 " is give n by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "rsolve(k(n+1)=(n+1)/(n+2)*k (n),k(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%\"kG6#\"\"!\"\"\",&% \"nG\"\"\"F+F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Example 2. 2" }{MPLTEXT 1 0 0 "" }}}{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+!\" \"F1F,F,-F(6#,&F+F,F,F,F,F1*(%&alphaGF,)F4F-\"\"\"-F(6#F+F,F,\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "REtoDE(RE,p(n),x);" }} {PARA 8 "" 1 "" {TEXT -1 92 "Error, (in REtoDE) this recurrence equati on has no classical orthogonal polynomial solutions" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Here we need to specify the parameter " } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "strict:=false:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "REtoD E(RE,p(n),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#(%DWarning:~parameter s~have~the~valuesG<(/%\"cGF'/%\"aG\"\"!/%\"eGF*/%\"bG,$F'\"\"#/%&alpha G#\"\"\"\"\"%/%\"dG,$F'!\"%" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%/,(*& ,&%\"xG\"\"#\"\"\"F*F*-%%diffG6$-%\"pG6$%\"nGF(-%\"$G6$F(F)F*#F*F)*&F( F*-F,6$F.F(F*!\"#*&F1F*F.F*F)\"\"!7$/%\"IG7$#!\"\"F)%)infinityG/-%$rho G6#F(,$-%$expG6#,$F(F9F)/*&&%\"kG6#,&F1F*F*F*\"\"\"&FO6#F1!\"\"F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "classical discrete orthogonal poly nomial solutions" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Example 3.1, same recurrence equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "REtodiscreteDE(RE,p(n),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%DWarning:~parameters~have~the~valuesG<*/%\"bG,&%\"dG#! \"\"\"\"#*&%\"fG\"\"\"F(F.F)/%\"cG,*F(#F.\"\"%*&)F-F+\"\"\"F(F6#F*F3*& %\"gGF.F(F6#F.F+*(F9F6F(F6F-F6F:/F-F-/F(F(/%\"aG\"\"!/%&alphaG,$*&,&F* F.*$F5F6F.F6*$)F-\"\"#F6!\"\"F2/%\"eG,$F8F*/F9F9" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&/,(*&*&,(%\"fG\"\"\"*&F)F*%\"xGF*\"\"#!\"\"F*F*-%&Del taG6$-%&NablaG6$-%\"pG6$%\"nG,&F+F*%\"gGF*F,F,F*\"\"\"F)!\"\"#F*F-*&*& F,F;-F0F4F*F;,&F*F*F)F*Ff*x+g and has the given weight function " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 48 "(x) and term ratio for the leading coe fficients." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Example 3.2" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "strict: =true:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "RE:=(n+2+alpha)* (2+n)*(2*n+2)*(n-N+1)*p(n+2)+(3+2*n)*(-6*n*alpha-2*n^2*alpha-4*n^2*x-1 2*n*x+2*n^2*N+6*n*N+4*N-4*alpha-8*x)*p(n+1)-(1+n)*(n+1-alpha)*(2*n+4)* (n+N+2)*p(n)=0;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#REG/,(*,,(%\"nG \"\"\"\"\"#F*%&alphaGF*F*,&F)F*F+F*F*,&F)F+F+F*F*,(F)F*%\"NG!\"\"F*F*F *-%\"pG6#F-F*F**(,&\"\"$F*F)F+F*,4*&F)F*F,F*!\"'*&F,\"\"\")F)F+F " 0 "" {MPLTEXT 1 0 26 "REt odiscreteDE(RE,p(n),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%DWarning:~ parameters~have~the~valuesG<'/%\"eG,(*&%\"NG\"\"\"%\"aGF*!\"\"*&%&alph aGF*F+\"\"\"\"\"#*(F.F/F+F/F)F/F*/%\"cG,&F-F,F1F,/F+F+/%\"dG,$F+F0/%\" bG,(F+F,F-F*F(F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%DWarning:~paramet ers~have~the~valuesG<'/%\"cG\"\"!/%\"eG,&*&%\"NG\"\"\"%\"aGF-!\"\"*(%& alphaGF-F.\"\"\"F,F2F//F.F./%\"dG,$F.\"\"#/%\"bG,(F.F/*&F1F2F.F2F-F+F/ " }}{PARA 6 "" 1 "" {TEXT -1 32 "Warning: several solutions found" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7%7%/,(*(,(%\"xG\"\"\"!\"\"F*%\"NGF+F* ,&F)F*%&alphaGF*F*-%&DeltaG6$-%&NablaG6$-%\"pG6$%\"nGF)F)F)F*F**&,*F) \"\"#F,F+F.F;*&F.F*F,F*F*F*-F0F4F*F**(F8F*,&F8F*F*F*F*F5F*F+\"\"!7$/-% &sigmaG6#F)*&F(\"\"\"F-FG/,&FCF*-%$tauGFEF**&,&F)F*F*F*F*,(F)F*F,F+F.F *F*/-%$rhoG6#F)-%*HypertermG6&7%F*,&F.F*F,F+F*7$,$F,F+,&F.F*F*F*F*F)7% /,(*(F)F*,*F)F*F+F*F,F+F.F*F*F/FGF**&,(F)!\"#F,F*FF+F@7$/ FC*&F)FGFinFG/FI*&,&F)F*F,F+F*,(F)F*F*F*F.F*F*/-FQ6#F)-FT6&FX7#FWF*F)/ *&&%\"kG6#,&F8F*F*F*FG&F]p6#F8!\"\",$*&,&F8F;F*F*FG*&,&F8F*F,F+\"\"\", (F8F*F.F*F*F*\"\"\"FbpF;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 246 "This result is interpreted as follows: The recurrence equation RE has two \+ different classical discrete orthogonal polynomial solutions which sat isfy the above difference equations. The resulting polynomial systems \+ have the given weight functions " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" } {TEXT -1 212 "(x) and term ratio for the leading coefficients. Note th at Hyperterm(upper,lower,z,x) denotes the hypergeometric term (= summa nd) of the hypergeometric function hypergeom(upper,lower,z) with summa tion variable x." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "classical q-o rthogonal polynomial solutions" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 11 "Example 4.1" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "q-Hermite" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "RE :=p(n+2)-x*p(n+1)+alpha*q^n*(q^(n+1)-1)*p(n)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,(-%\"pG6#,&%\"nG\"\"\"\"\"#F,F,*&%\"xGF,-F(6#,& F+F,F,F,F,!\"\"**%&alphaGF,)%\"qGF+F,,&)F7F2F,F3F,F,-F(6#F+F,F,\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "strict:=true:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "REtoqDE(RE,p(n),q,x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%DWarning:~parameters~have~the~valuesG<'/%\"eG\" \"!/%\"aG,&*&%\"dG\"\"\"%\"qGF-!\"\"F,F-/%\"cG,&*(%&alphaGF-F.\"\"\"F, F5F/*&F4F5F,F5F-/F,F,/%\"bGF'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%/,( *&,&*$)%\"xG\"\"#\"\"\"\"\"\"%&alphaGF-F--%#DqG6%-F06%-%\"pG6$%\"nGF** &F,F,%\"qG!\"\"F*F9F*F-F-*&*&F*F--F06%F4F9F*F-F,,&F9F-!\"\"F-F:F@*&*(F 9F-,&F@F-)F9F7F-F-F4F-F,*&)F?\"\"#F,FD\"\"\"F:F-\"\"!/*&-%$rhoG6#*&F9F ,F*F,F,-FM6#F*F:*&F.F,,&*&)F9F+F,F)F,F-F.F-F:/*&&%\"kG6#,&F7F-F-F-F,&F Y6#F7F:F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "This result is inte rpreted as follows: The recurrence equation RE has a q-classical ortho gonal polynomial solution which satisfies the q-difference equation" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "(x^2+alpha)*Dq(Dq(p(n,x),1 /q,x),q,x)-x*Dq(p(n,x),q,x)/(q-1)+q*(-1+q^n)*p(n,x)/((q-1)^2*q^n) = 0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&*$)%\"xG\"\"#\"\"\"\"\"\"%& alphaGF,F,-%#DqG6%-F/6%-%\"pG6$%\"nGF)*&F+F+%\"qG!\"\"F)F8F)F,F,*&*&F) F,-F/6%F3F8F)F,F+,&F8F,!\"\"F,F9F?*&*(F8F,,&F?F,)F8F6F,F,F3F,F+*&)F>\" \"#F+FC\"\"\"F9F,\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "and its density " }{XPPEDIT 18 0 "rho(x);" "6#-%$rhoG6#%\"xG" }{TEXT -1 59 " \+ has the given term ratio. Its standardization is such that" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "k[n+1]/k[n]=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"kG6#,&%\"nG\"\"\"F*F*\"\"\"&F &6#F)!\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "hence the polynom ials are monic or" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "k[n]=k[0];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"kG6#%\"nG&F%6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "If we consider arbitrary translates, we g et" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "st rict:=false:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "REtoqDE(RE, p(n),q,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%DWarning:~parameters~ha ve~the~valuesG<*/%\"eG\"\"!/%\"gGF'/%\"cGF'/%\"dGF-/%\"aGF//%&alphaGF1 /%\"bGF'/%\"fGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%DWarning:~paramet ers~have~the~valuesG<*/%\"eG\"\"!/%\"gGF'/%\"cGF'/%\"fGF-/%\"dGF//%\"a GF1/%\"bGF'/%&alphaGF'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%DWarning:~p arameters~have~the~valuesG<*/%\"eG\"\"!/%\"gGF'/%\"cG,$**%&alphaG\"\" \")%\"fG\"\"#\"\"\"%\"dGF/,&%\"qGF/!\"\"F/F/F7/F1F1/F4F4/%\"aG,&*&F4F3 F6F/F7F4F//F.F./%\"bGF'" }}{PARA 6 "" 1 "" {TEXT -1 32 "Warning: sever al solutions found" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%7$/,(*&)%\"xG \"\"#\"\"\"-%#DqG6%-F-6%-%\"pG6$%\"nG*&%\"fG\"\"\"F)F7*&F+F+%\"qG!\"\" F)F9F)F7F7*&*(%\"dGF7F)F+-F-6%F1F9F)F7F+*&F6\"\"\"%\"aG\"\"\"F:F7*&*(, &!\"\"F7)F9F4F7F7,**(F=F+FHF7F9F7F7*&F=F+FHF+FG*&FBF7FHF+F7*&FBF+F9F+F GF7F1F7F+**FH\"\"\"),&F9F7FGF7\"\"#F+)F6\"\"#F+FB\"\"\"F:FG\"\"!/*&-%$ rhoG6#*&F9F+F)F+F+-FZ6#F)F:*&,(*&F=F+F9F+F7F=FGFBF7F+*&FB\"\"\")F9\"\" #F+F:7$/,(*&,&*$F(F+F7%&alphaGF7F7F,F+F7*&*&F)F+F>F+F+*&F6\"\"\"FQ\"\" \"F:FG*&*(F9F+FFF+F1F+F+*()FQ\"\"#F+FH\"\"\")F6\"\"#F+F:F7FV/FX*&*&Ffo F+)F6F*F+F+,&*&)F9F*F+F(F+F7*&FfoF7FgpF+F7F:/*&&%\"kG6#,&F4F7F7F7F+&F_ q6#F4F:*&F+F+F6F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "This result is interpreted as follows: The recurrence equation RE has a q-classic al orthogonal polynomial solution which satisfies the q-difference equ ation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "(x^2+alpha)*Dq(Dq (p(n,f*x),1/q,x),q,x)-x*Dq(p(n,f*x),q,x)/(f*(q-1))+q*(-1+q^n)*p(n,f*x) /((q-1)^2*q^n*f^2) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&*$)% \"xG\"\"#\"\"\"\"\"\"%&alphaGF,F,-%#DqG6%-F/6%-%\"pG6$%\"nG*&%\"fGF,F) F,*&F+F+%\"qG!\"\"F)F:F)F,F,*&*&F)F+-F/6%F3F:F)F,F+*&F8\"\"\",&F:F,!\" \"F,\"\"\"F;FC*&*(F:F,,&FCF,)F:F6F,F,F3F,F+*()FB\"\"#F+FH\"\"\")F8\"\" #F+F;F,\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "and its density \+ " }{XPPEDIT 18 0 "rho(x);" "6#-%$rhoG6#%\"xG" }{TEXT -1 59 " has the g iven term ratio. Its standardization is such that" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "k[n+1]/k[n]=1/f;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"kG6#,&%\"nG\"\"\"F*F*\"\"\"&F&6#F)!\"\"*&F+F+%\" fGF." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "The first solution is spu rious (c=0) and has no q-orthogonal solution." }{MPLTEXT 1 0 0 "" }}}} {MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }