{VERSION 3 0 "IBM INTEL NT" "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 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 "Times" 1 14 0 0 0 0 2 2 2 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 }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 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read `c:/koepf/maple/buch/hsum.mpl` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZCopyright~1998~~Wolfram~Koepf,~ Konrad-Zuse-Zentrum~BerlinG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "sumrecursion(exp(-t)/n*(-1)^k/(k-1)!*binomial(n,k)*(2*t)^k,k,F(n)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\"#F(F(-%\"FG 6#F&F(F(*&,(!\"\"F(F'F/%\"tGF(F(-F+6#,&F'F(F(F(F(F)*&F'F(-F+6#F'F(F(\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "rodriguesrecursion(t *exp(t)/n!,exp(-2*t)*t^(n-1),t,F(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\"#F(F(-%\"FG6#F&F(F(*&,(!\"\"F(F'F/%\"tGF(F(-F +6#,&F'F(F(F(F(F)*&F'F(-F+6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "GFrecursion(exp(-t*(1+z)/(1-z)),1,z,F(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\"#F(F(-%\"FG6#F&F(F(*&, (!\"\"F(F'F/%\"tGF(F(-F+6#,&F'F(F(F(F(F)*&F'F(-F+6#F'F(F(\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sumrecursion(n!*(-1)^k*(2*x) ^(n-2*k)/(n-2*k)!/k!,k,H(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-% \"HG6#,&%\"nG\"\"\"\"\"#F*F**&%\"xGF*-F&6#,&F)F*F*F*F*!\"#*&F0F*-F&6#F )F*F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "rodriguesrecu rsion((-1)^n*exp(x^2),exp(-x^2),x,H(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%\"HG6#,&%\"nG\"\"\"\"\"#F*F**&%\"xGF*-F&6#,&F)F*F*F*F*!\"#* &F0F*-F&6#F)F*F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "su mrecursion(n!*(-1)^k*(2*x)^(n-2*k)/(n-2*k)!/k!*z^n/n!,n,S(k));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&%\"kG\"\"\"F(F(F(-%\"SG6#F&F(F( *&)%\"zG\"\"#\"\"\"-F*6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "sum(subs(k=0,n!*(-1)^k*(2*x)^(n-2*k)/(n-2*k)!/k!*z^n/ n!),n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$*&% \"xG\"\"\"%\"zGF)\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "G Frecursion(exp(2*x*z-z^2),1/n!,z,H(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%\"HG6#,&%\"nG\"\"\"\"\"#F*F**&%\"xGF*-F&6#,&F)F*F*F*F*!\"#* &F0F*-F&6#F)F*F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "su mrecursion(binomial(n+alpha,n)*\nhyperterm([-n,n+alpha+beta+1],[alpha+ 1],(1-x)/2,k),k,P(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(**,&%\"nG \"\"\"\"\"#F(F(,*%%betaGF(F)F(%&alphaGF(F'F)F(,*F'F(F)F(F,F(F+F(F(-%\" PG6#F&F(F)*(,*F'F)\"\"$F(F+F(F,F(F(,:*$)F+F)\"\"\"!\"\"*&%\"xGF(F6F7F( *&F+F(F:F7\"\"'*(F+F7F:F7F'F(\"\"%*(F:F7F,F(F+F7F)*&)F'F)F7F:F7F>*&F,F 7F:F7F<*$)F,F)F7F(*(F,F7F:F7F'F7F>*&F'F7F:F7\"#7F:\"\")*&F:F7FDF7F(F(- F/6#,&F'F(F(F(F(F8**,(F'F(F,F(F(F(F(,(F+F(F'F(F(F(F(,*F+F(F,F(F'F)F>F( F(-F/6#F'F(F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "rodr iguesrecursion((-1)^n/(2^n*n!)*(1-x)^(-alpha)*(1+x)^(-beta),\n(1-x)^al pha*(1+x)^beta*(1-x^2)^n,x,P(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/ ,(**,&%\"nG\"\"\"\"\"#F(F(,*%%betaGF(F)F(%&alphaGF(F'F)F(,*F'F(F)F(F,F (F+F(F(-%\"PG6#F&F(F)*(,*F'F)\"\"$F(F+F(F,F(F(,:*$)F+F)\"\"\"!\"\"*&% \"xGF(F6F7F(*&F+F(F:F7\"\"'*(F+F7F:F7F'F(\"\"%*(F:F7F,F(F+F7F)*&)F'F)F 7F:F7F>*&F,F7F:F7F<*$)F,F)F7F(*(F,F7F:F7F'F7F>*&F'F7F:F7\"#7F:\"\")*&F :F7FDF7F(F(-F/6#,&F'F(F(F(F(F8**,(F'F(F,F(F(F(F(,(F+F(F'F(F(F(F(,*F+F( F,F(F'F)F>F(F(-F/6#F'F(F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "simpcomb(Sumtohyper(pochhammer(2*nu,n)/pochhammer(nu+1/2,n)* \nsubs(alpha=nu-1/2,beta=nu-1/2,binomial(n+alpha,n)*\nhyperterm([-n,n+ alpha+beta+1],[alpha+1],(1-x)/2,k)),k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&-%&GAMMAG6#,&%\"nG\"\"\"%#nuG\"\"#F*-%*HypergeomG6%7$,$F)!\" \"F(7#,&F+F*#F*F,F*,&F5F*%\"xG#F2F,F*\"\"\"*&-F&6#,$F+F,\"\"\"-F&6#,&F )F*F*F*\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "GFrec ursion((1-2*x*z+z^2)^(-nu),1,z,C(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\"#F(F(-%\"CG6#F&F(F(*(%\"xGF(,(F'F(F(F(%#nuGF( F(-F+6#,&F'F(F(F(F(!\"#*&,&F'F(F0F)F(-F+6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "sumrecursion(pochhammer(2*nu,n)/po chhammer(nu+1/2,n)*\nsubs(alpha=nu-1/2,beta=nu-1/2,binomial(n+alpha,n) *\nhyperterm([-n,n+alpha+beta+1],[alpha+1],(1-x)/2,k)),k,C(n));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\"#F(F(-%\"CG6#F&F (F(*(%\"xGF(,(F'F(F(F(%#nuGF(F(-F+6#,&F'F(F(F(F(!\"#*&,&F'F(F0F)F(-F+6 #F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "sumrecursi on(pochhammer(2*nu,n)/pochhammer(nu+1/2,n)*\nsubs(alpha=nu-1/2,beta=nu -1/2,binomial(n+alpha,n)*\nhyperterm([-n,n+alpha+beta+1],[alpha+1],(1- x)/2,k))*z^n,n,S(k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(),&%\"zG \"\"\"!\"\"F)\"\"#\"\"\",&%\"kGF)F)F)F)-%\"SG6#F-F)F***F(F),&%#nuGF)F. F)F),&F*F)%\"xGF)F)-F06#F.F)F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "init:=sum(binomial(2*nu+n-1,n)*z^n,n=0..infinity);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%initG),&\"\"\"F'%\"zG!\"\",$%#nuG! \"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "result:=init*(1+2*z* (1-x)/(1-z)^2)^(-nu);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'resultG*&) ,&\"\"\"F(%\"zG!\"\",$%#nuG!\"#F(),&F(F(*&*&F)F(,&F(F(%\"xGF*F(\"\"\"* $)F'\"\"#F4!\"\"\"\"#,$F,F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(result);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&),&\"\"\" F&%\"zG!\"\",$%#nuG!\"#F&),$*&,(F(F&*&%\"xGF&F'F&\"\"#*$)F'F2\"\"\"F(F 5*$),&F'F&F(F&\"\"#F5!\"\"F(,$F*F(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "GFrecursion(1/(1-z-z^2),1,z,F(n));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,(-%\"FG6#%\"nG\"\"\"-F&6#,&F(F)F)F)F)-F&6#,&F(F)\" \"#F)!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "sumrecur sion(subs(alpha=alpha-n,\nbinomial(n+alpha,n)*hyperterm([-n],[alpha+1] ,x,k)),k,L(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\" \"\"#F(F(-%\"LG6#F&F(F(*&,*%\"xGF(%&alphaG!\"\"F(F(F'F(F(-F+6#,&F'F(F( F(F(F(*&F/F(-F+6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "GFrecursion(exp(-x*z)*(1+z)^alpha,1,z,L(n));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\"#F(F(-%\"LG6#F&F(F(*&,*%\"xGF(% &alphaG!\"\"F(F(F'F(F(-F+6#,&F'F(F(F(F(F(*&F/F(-F+6#F'F(F(\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "sumrecursion(subs(alpha=alph a-n,\nbinomial(n+alpha,n)*hyperterm([-n],[alpha+1],x,k))*z^n,n,S(k)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&%\"kG\"\"\"F(F(F(-%\"SG6#F& F(F(*(%\"xGF(%\"zGF(-F*6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "init:=sum(subs(alpha=alpha-n,binomial(n+alpha,n))*z^n ,n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%initG),&\"\"\"F '%\"zGF'%&alphaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "CTrecur sion((z+x+1/z)^n,z,S(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*&,&% \"nG\"\"\"\"\"#F(F(-%\"SG6#F&F(F(*(%\"xGF(,&F'F)\"\"$F(F(-F+6#,&F'F(F( F(F(!\"\"**-F+6#F'F(,&F.F(!\"#F(F(,&F.F(F)F(F(F3F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "CTdiffeq((z+x+1/z)^n,z,S(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(,&%\"xG\"\"\"!\"#F(F(,&F'F(\"\"#F (F(-%%diffG6$-%\"SG6#F'-%\"$G6$F'F+F(F(*(F'F(,&%\"nGF+!\"\"F(F(-F-6$F/ F'F(F8*&F/F()F7F+\"\"\"F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "sumrecursion(binomial(n,k)*binomial(-n-1,k)*((1-x)/2)^k,k,P(n) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\"#F(F(-%\"P G6#F&F(F(*(,&F'F)\"\"$F(F(%\"xGF(-F+6#,&F'F(F(F(F(!\"\"*&F3F(-F+6#F'F( F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "CTrecursion((((x +z)^2-1)/(2*z))^n,z,P(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&% \"nG\"\"\"\"\"#F(F(-%\"PG6#F&F(F(*(,&F'F)\"\"$F(F(%\"xGF(-F+6#,&F'F(F( F(F(!\"\"*&F3F(-F+6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sumdiffeq(binomial(n,k)*binomial(-n-1,k)*((1-x)/2)^k,k,P(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(,&%\"xG\"\"\"!\"\"F(F(,&F(F(F 'F(F(-%%diffG6$-%\"PG6#F'-%\"$G6$F'\"\"#F(F)*&F'F(-F,6$F.F'F(!\"#*(F.F (%\"nGF(,&F:F(F(F(F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "CTdiffeq((((x+z)^2-1)/(2*z))^n,z,P(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(,&%\"xG\"\"\"!\"\"F(F(,&F(F(F'F(F(-%%diffG6$-%\"PG 6#F'-%\"$G6$F'\"\"#F(F)*&F'F(-F,6$F.F'F(!\"#*(F.F(%\"nGF(,&F:F(F(F(F(F (\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "sumrecursion(bino mial(n,k)*binomial(-n-1,k)*((1-x)/2)^k,k,P(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"\"\"#F(F(-%\"PG6#F&F(F(*(,&F'F)\"\"$F (F(%\"xGF(-F+6#,&F'F(F(F(F(!\"\"*&F3F(-F+6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "sumrecursion(x^n*hyperterm([-n/2,(1 -n)/2],[1],1-1/x^2,k),k,P(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(* &,&%\"nG\"\"\"\"\"#F(F(-%\"PG6#F&F(F(*(,&F'F)\"\"$F(F(%\"xGF(-F+6#,&F' F(F(F(F(!\"\"*&F3F(-F+6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(subs(n=0,x^n*hypergeom([-n/2,(1-n)/2],[1],1- 1/x^2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(subs(n=1,x^n*hypergeom([-n/2,(1- n)/2],[1],1-1/x^2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "sumrecursion(x^n*hyperterm([ -n/2,(1-n)/2],[1],1-1/x^2,k)/n!*z^n,n,S(k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&),&%\"kG\"\"\"F)F)\"\"#\"\"\"-%\"SG6#F'F)!\"%**,&% \"xGF)!\"\"F)F),&F)F)F2F)F))%\"zGF*F+-F-6#F(F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "init:=sum(x^n/n!*z^n,n=0..infinity) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%initG-%$expG6#*&%\"xG\"\"\"%\" zGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "init*sum(hyperterm( [],[1],(x^2-1)/4*z^2,k),k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#*&%\"xG\"\"\"%\"zGF)F)-%(BesselIG6$\"\"!*$-%%sqrtG6#* &)F*\"\"#\"\"\",&*$)F(F5F6F)!\"\"F)F)F6F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }