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-1 213 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 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle 11" -1 214 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle12" -1 215 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle13" -1 216 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_psty le14" -1 217 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle15" -1 218 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle16" -1 219 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 } {PSTYLE "_pstyle17" -1 220 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle19" -1 222 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle20" -1 223 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 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle21" -1 224 1 {CSTYLE "" -1 -1 "Time s" 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 "_pstyle10" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 212 "" 0 "" {TEXT 222 32 "Some Identities Chapter 1 of A=B" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 223 42 "A trigonometrical identity sinus of angles" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 23 "f:=arcsin(x)+arcsin(y);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>% \"fG,&-%'arcsinG6#%\"xG\"\"\"-F'6#%\"yGF*" }{TEXT 225 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 41 "g:=arcsin(x*sqrt(1-y**2) +y*sqrt(1-x**2));" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"gG-%'arcsinG 6#,&*&%\"xG\"\"\",&F+F+*$)%\"yG\"\"#F+!\"\"#F+F0F+*&F/F+,&F+F+*$)F*F0F +F1F2F+" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 14 "f1:=diff(f,x);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%#f1G*&\" \"\"F&*$,&F&F&*$)%\"xG\"\"#F&!\"\"#F&F,F-" }{TEXT 225 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 14 "g1:=diff(g,x);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%#g1G*&,&*$,&\"\"\"F)*$)%\"yG\"\"#F)!\"\"# F)F-F)*(F,F),&F)F)*$)%\"xGF-F)F.#F.F-F4F)F.F),&F)F)*$),&*&F4F)F(F/F)*& F,F)F1F/F)F-F)F.F5" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 38 "normal(simplify(expand(g1**2))-f1**2);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"!" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "" 0 "" {TEXT 223 40 "Fibonacci numbers and Cassini's identity" }} }{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 90 "F:=proc(n) local phi,ps i;;phi:=(1+sqrt(5))/2;psi:=(1-sqrt(5))/2;(phi^n-psi^n)/sqrt(5);end;" } }{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"nG6$%$phiG%$psiG6\"F+C% >8$,&#\"\"\"\"\"#F1*&F0F1-%%sqrtG6#\"\"&F1F1>8%,&F0F1F3!\"\"*&,&)F.9$F 1)F9F?F;F1F4F;F+F+F+" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 28 "seq(simplify(F(k)),k=0..10);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6-\"\"!\"\"\"F$\"\"#\"\"$\"\"&\"\")\"#8\"#@\"#M\"#b" } {TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 26 "Cas: =F(n+1)*F(n-1)-F(n)^2:" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 27 "Cas:=expand(simplify(Cas));" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>% $CasG)!\"\"%\"nG" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 30 "numer(Cas)/expand(denom(Cas));" }}{PARA 214 "" 1 " " {XPPMATH 20 "6#)!\"\"%\"nG" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 " " 0 "" {TEXT 223 38 "Elliptic integrals and theta functions" }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 12 "assume(k>0);" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 48 "F:=z->int(1/sqrt(1-t^2)/sqrt(1- k^2*t^2),t=0..z);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"zG6 \"6$%)operatorG%&arrowGF(-%$intG6$*&\"\"\"F0*&-%%sqrtG6#,&F0F0*$)%\"tG \"\"#F0!\"\"F0-F36#,&F0F0*&)%\"kGF9F0F7F0F:F0F:/F8;\"\"!9$F(F(F(" } {TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 53 "simp lify(diff(F(z),z)-1/sqrt(1-z^2)/sqrt(1-k^2*z^2));" }}{PARA 214 "" 1 " " {XPPMATH 20 "6#\"\"!" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 " " {MPLTEXT 1 224 18 "lef:=x->F(x)+F(y);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%$lefGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%\"FG6#9$\"\"\"-F.6 #%\"yGF1F(F(F(" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 86 "rig:=F((x*sqrt(1-y^2)*sqrt(1-k^2*y^2)+y*sqrt(1-x^2) *sqrt(1-k^2*x^2))/(1-k^2*x^2*y^2)):" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 21 "lef1:=diff(lef(x),x);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%%lef1G*&\"\"\"F&*&,&F&F&*$)%\"xG\"\"#F&!\"\"#F&F,,&F& F&*&)%#k|irGF,F&F*F&F-F.F-" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> \+ " 0 "" {MPLTEXT 1 224 18 "rig1:=diff(rig,x):" }}}{EXCHG {PARA 213 "> \+ " 0 "" {MPLTEXT 1 224 24 "simplify(lef1^2-rig1^2);" }}{PARA 214 "" 1 " " {XPPMATH 20 "6#\"\"!" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 " " {MPLTEXT 1 224 36 "g:=z->1/sqrt(1-z^2)/sqrt(1-k^2*z^2):" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 24 "rig2:=subs(D(F)=g,rig1):" }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 26 "gu:=normal((rig2/lef1)^2 ):" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 36 "expand(numer(gu)) /expand(denom(gu));" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#%#guG" }{TEXT 225 2 " " }}}{EXCHG {PARA 215 "" 0 "" {TEXT -1 0 "" }}{PARA 215 "" 0 "" {TEXT 226 9 "Chapter 2" }}{PARA 213 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 31 "with(SumTools[Hypergeome tric]):" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 17 "T:=binomial( n,k);" }}{PARA 213 "> " 0 "" {MPLTEXT 1 224 29 "WZ:= WZMethod(T,2^n,n, k,'R');" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"TG-%)binomialG6$%\"nG% \"kG" }{TEXT 225 2 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%#WZG7$*&- %)binomialG6$%\"nG%\"kG\"\"\")\"\"#F*!\"\"*(F+F,,(F*F,F,F,*&F.F,F+F,F/ F/,&*&-F(6$,&F*F,F,F,F+F,)F.F7F/F,F&F/F," }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 2 "R;" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#!\"\"%\"kG\"\"\",(%\"nGF(F'F&F(F(F&F&" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 38 "T:=binomia l(n,k)^2;r:=binomial(2*n,n);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"T G*$)-%)binomialG6$%\"nG%\"kG\"\"#\"\"\"" }{TEXT 225 2 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"rG-%)binomialG6$,$*&\"\"#\"\"\"%\"nGF+F +F," }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 28 "WZ := WZMethod(T,r,n,k,'R');" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#> %#WZG7$*&-%)binomialG6$%\"nG%\"kG\"\"#-F(6$,$*&F,\"\"\"F*F1F1F*!\"\"** ,(*&\"\"$F1F*F1F1F6F1*&F,F1F+F1F2F1F+F,,4*&F6F1)F*F6F1F1*&\"\"(F1)F*F, F1F1*&\"\"&F1F*F1F1F1F1*(\"\")F1F=F1F+F1F2*(\"\"%F1)F+F,F1F*F1F1*(\"#7 F1F*F1F+F1F2*&F,F1FDF1F1*&FCF1F+F1F2F2,&*&-F(6$,&F*F1F1F1F+F,-F(6$,&F0 F1F,F1FMF2F1F&F2F1" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 2 "R;" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,$*,\"\"#!\" \"%\"kGF%,(*&\"\"$\"\"\"%\"nGF+F+F*F+*&F%F+F'F+F&F+,(F,F+F'F&F+F+!\"#, &*&F%F+F,F+F+F+F+F&F&" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 " " {MPLTEXT 1 224 83 "T:=(n,k,i,j)->(n-i)!*(n-j)!*(i-1)!*(j-1)!/((n-1)! *(k-1)!*(n-i-j+k)!*(i-k)!*(j-k)!);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6 #>%\"TGf*6&%\"nG%\"kG%\"iG%\"jG6\"6$%)operatorG%&arrowGF+*4-%*factoria lG6#,&9$\"\"\"9&!\"\"F5-F16#,&F4F59'F7F5-F16#,&F6F5F5F7F5-F16#,&F;F5F5 F7F5-F16#,&F4F5F5F7F7-F16#,&9%F5F5F7F7-F16#,*F4F5F6F7F;F7FHF5F7-F16#,& F6F5FHF7F7-F16#,&F;F5FHF7F7F+F+F+" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 5 "r:=1;" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 37 "WZ := WZMethod(T(n,k,i,j),r,n,k,'R');" }}{PARA 216 "" 1 "" {XPPMATH 20 "6#>%#WZG7$*4-%*factorialG6#,&%\"nG\"\"\"%\"iG!\" \"F,-F(6#,&F+F,%\"jGF.F,-F(6#,&F-F,F,F.F,-F(6#,&F2F,F,F.F,-F(6#,&F+F,F ,F.F.-F(6#,&%\"kGF,F,F.F.-F(6#,*F+F,F-F.F2F.F?F,F.-F(6#,&F-F,F?F.F.-F( 6#,&F2F,F?F.F.,$**,,F+F.F,F.F-F,F2F,F?F.F,F>F,,.F+F,F,F,F2F.F-F.*&F-F, F2F,F,*&F+F,F?F,F.F.,&*4-F(6#,(F+F,F,F,F-F.F,-F(6#,(F+F,F,F,F2F.F,F3F, F6F,-F(6#F+F.F " 0 "" {MPLTEXT 1 224 2 "R;" } }{PARA 214 "" 1 "" {XPPMATH 20 "6#*&,&%\"kG\"\"\"F&!\"\"F&%\"nGF'" } {TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 21 "simp lify(T(n,k,i,j));" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#*4-%&GAMMAG6#,(% \"nG\"\"\"F)F)%\"iG!\"\"F)-F%6#,(F(F)F)F)%\"jGF+F)-F%6#F*F)-F%6#F/F)-F %6#F(F+-F%6#%\"kGF+-F%6#,,F(F)F)F)F*F+F/F+F8F)F+-F%6#,(F)F)F*F)F8F+F+- F%6#,(F)F)F/F)F8F+F+" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 11 "T(i,j,i,j);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\" \"\"" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 186 "G:=(n,k)->-(-n-1+i+j-k)*(k-1)/(n+1-j-i+i*j-n*k)*((n+1-i)!*(n+1-j) !*(i-1)!*(j-1)!/n!/(k-1)!/(n+1-i-j+k)!/(i-k)!/(j-k)!-(n-i)!*(n-j)!*(i- 1)!*(j-1)!/(n-1)!/(k-1)!/(n-i-j+k)!/(i-k)!/(j-k)!):" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 50 "simplify(T(n+1,k,i,j)-T(n,k,i,j)-G(n, k+1)+G(n,k));" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"!" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 33 "f:=n->(3*n+1)!*( 2*n-5)!/(n+2)!^2;" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"nG6 \"6$%)operatorG%&arrowGF(*(-%*factorialG6#,&*&\"\"$\"\"\"9$F3F3F3F3F3- F.6#,&*&\"\"#F3F4F3F3\"\"&!\"\"F3-F.6#,&F4F3F9F3!\"#F(F(F(" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 37 "u:=simplif y(add(f(n-k)/f(n),k=0..3));" }}{PARA 216 "" 1 "" {XPPMATH 20 "6#>%\"uG ,$*B\"$;#!\"\",B\"*+3`,\"\"\"\"*&\",glafc\"F+)%\"nG\"\"#F+F+*&\"+SG[nF F+F/F+F(*&\")G4oRF+)F/\"#9F+F(*&\"(7(f7F+)F/\"#:F+F+*&\"+/8DcYF+)F/\"# 7F+F(*&\"*_r?f&F+)F/\"#8F+F+*&\",i)[]6AF+)F/\"\"$F+F(*&\"-Mp%3].$F+)F/ \"\"&F+F+*&\"-n![VCq&F+)F/\"\"'F+F(*&\"-y)))\\IW'F+)F/\"\"(F+F+*&\"-yv +hr[F+)F/\"\")F+F(*&\"-kb+FsDF+)F/\"\"*F+F+*&\",cl'**H'*F+)F/\"#5F+F(* &\",)G.^ZDF+)F/\"#6F+F+*&\",,6vXN'F+)F/\"\"%F+F(F+,&F/F+FFF(F(,&*&F0F+ F/F+F+FJF(F(F/F(,&F/F+F`oF(F(,&FcoF+FRF(F(,&*&FFF+F/F+F+F`oF(F(,&F/F+F +F(F(,&FgoF+F0F(F(,&F/F+FJF(F(,&FcoF+FZF(F(,&FgoF+FRF(F(,&F/F+F0F(F(,& FgoF+FJF(F(,&*&FZF+F.F+F+F+F(F(F+" }{TEXT 227 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 20 "v:=sort(numer(u),n);" }}{PARA 216 "" 1 "" {XPPMATH 20 "6#>%\"vG,B*&\"(7(f7\"\"\")%\"nG\"#:F(F(*&\")G4oRF()F *\"#9F(!\"\"*&\"*_r?f&F()F*\"#8F(F(*&\"+/8DcYF()F*\"#7F(F0*&\",)G.^ZDF ()F*\"#6F(F(*&\",cl'**H'*F()F*\"#5F(F0*&\"-kb+FsDF()F*\"\"*F(F(*&\"-yv +hr[F()F*\"\")F(F0*&\"-y)))\\IW'F()F*\"\"(F(F(*&\"-n![VCq&F()F*\"\"'F( F0*&\"-Mp%3].$F()F*\"\"&F(F(*&\",,6vXN'F()F*\"\"%F(F0*&\",i)[]6AF()F* \"\"$F(F0*&\",glafc\"F()F*\"\"#F(F(*&\"+SG[nFF(F*F(F0\"*+3`,\"F(" } {TEXT 227 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 28 "w:=s ort(expand(denom(u)),n);" }}{PARA 216 "" 1 "" {XPPMATH 20 "6#>%\"wG,@* &\"(7(f7\"\"\")%\"nG\"#:F(F(*&\")G4oRF()F*\"#9F(!\"\"*&\"*_r?f&F()F*\" #8F(F(*&\"+oHEcYF()F*\"#7F(F0*&\",'*)f`ZDF()F*\"#6F(F(*&\",kQA-j*F()F* \"#5F(F0*&\"-wJdOsDF()F*\"\"*F(F(*&\"-%y6g<([F()F*\"\")F(F0*&\"-_p'RFW 'F()F*\"\"(F(F(*&\"-3?Vq+dF()F*\"\"'F(F0*&\"-sc " 0 "" {MPLTEXT 1 224 12 "sort(v-w,n);" }}{PARA 216 "" 1 "" {XPPMATH 20 "6#,< *&\"&k;\"\"\"\")%\"nG\"#7F&F&*&\"'3mDF&)F(\"#6F&!\"\"*&\"(3tD#F&)F(\"# 5F&F&*&\"(7wc*F&)F(\"\"*F&F.*&\")1U+:F&)F(\"\")F&F&*&\")E>-JF&)F(\"\"( F&F&*&\"*fg\"R " 0 "" {MPLTEXT 1 224 32 "T:=(-1)^k*binomial(n,k)*x/(x+k);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"TG**)!\"\"%\"kG\"\"\"-%)binomialG6$%\"nGF(F)%\"x GF),&F.F)F(F)F'" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 21 "r:=1/binomial(x+n,n);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"rG*&\"\"\"F&-%)binomialG6$,&%\"xGF&%\"nGF&F,!\"\"" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 28 "WZ \+ := WZMethod(T,r,n,k,'R');" }}{PARA 216 "" 1 "" {XPPMATH 20 "6#>%#WZG7$ *,)!\"\"%\"kG\"\"\"-%)binomialG6$%\"nGF)F*%\"xGF*,&F/F*F)F*F(-F,6$,&F/ F*F.F*F.F*,$*(F)F*,&F.F*F*F*F(,&*,F'F*-F,6$F6F)F*F/F*F0F(-F,6$,(F/F*F. F*F*F*F6F*F*F&F(F*F(" }{TEXT 227 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 2 "R;" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,$**%\"kG\"\" \",&%\"xGF&F%F&F&,&%\"nGF&F&F&!\"\",(F*F&F%F+F&F&F+F+" }{TEXT 225 2 " \+ " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 14 "sum(T,k=0..0);" }} {PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"\"" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 40 "r:='r';T:=binomial(n,k)*binomia l(x,k+r);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"rGF$" }{TEXT 225 2 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"TG*&-%)binomialG6$%\"nG%\"k G\"\"\"-F'6$%\"xG,&F*F+%\"rGF+F+" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 21 "f:=binomial(n+x,n+r);" }}{PARA 214 " " 1 "" {XPPMATH 20 "6#>%\"fG-%)binomialG6$,&%\"xG\"\"\"%\"nGF*,&F+F*% \"rGF*" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 28 "WZ := WZMethod(T,f,n,k,'R');" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%#WZG7$*(-%)binomialG6$%\"nG%\"kG\"\"\"-F(6$%\"xG,&F+F,%\"rGF,F,-F (6$,&F/F,F*F,,&F*F,F1F,!\"\"**F+F,F0F,,0*&F*F,F1F,F6F1F6*&F/F,F*F,F,*& F+F,F/F,F6*&F*F,F+F,F6F+F6F/F,F6,&*(-F(6$,&F*F,F,F,F+F,F-F,-F(6$,(F/F, F*F,F,F,,(F*F,F,F,F1F,F6F,F&F6F," }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 2 "R;" }}{PARA 214 "" 1 "" {XPPMATH 20 "6 #,$**%\"kG\"\"\",&F%F&%\"rGF&F&,(%\"nGF&F%!\"\"F&F&F+,(%\"xGF&F*F&F&F& F+F+" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 14 "sum(T,k=0..0);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#-%)binomialG6$% \"xG%\"rG" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 53 "T:=binomial(x+1,2*k+1)*binomial(x-2*k,n-k)*2^(2*k+1);" }} {PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"TG*(-%)binomialG6$,&%\"xG\"\"\"F+ F+,&*&\"\"#F+%\"kGF+F+F+F+F+-F'6$,&F*F+F-!\"\",&%\"nGF+F/F3F+)F.F,F+" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 25 "f:= binomial(2*x+2,2*n+1);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"fG-%)bi nomialG6$,&*&\"\"#\"\"\"%\"xGF+F+F*F+,&*&F*F+%\"nGF+F+F+F+" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 28 "WZ := WZMe thod(T,f,n,k,'R');" }}{PARA 216 "" 1 "" {XPPMATH 20 "6#>%#WZG7$**-%)bi nomialG6$,&%\"xG\"\"\"F,F,,&*&\"\"#F,%\"kGF,F,F,F,F,-F(6$,&F+F,F.!\"\" ,&%\"nGF,F0F4F,)F/F-F,-F(6$,&*&F/F,F+F,F,F/F,,&*&F/F,F6F,F,F,F,F4**F-F ,F0F,,.*(F/F,F+F,F6F,F,*(F/F,F0F,F+F,F4F;F,*&F/F,)F6F/F,F4F=F4*&\"\"$F ,F0F,F4F4,&**F'F,-F(6$F3,(F6F,F0F4F,F,F,F7F,-F(6$F:,&F=F,FEF,F4F,F&F4F ," }{TEXT 227 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 2 "R ;" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,$*.%\"kG\"\"\",&*&\"\"#F&F%F&F& F&F&F&,(%\"xGF&*&F)F&%\"nGF&!\"\"F&F.F&,(F-F&F%F.F&F&F.,&F+F&F-F.F.,(* &F)F&F+F&F&F&F&F,F.F.F." }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 14 "sum(T,k=0..0);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\",&%\"xGF&F&F&F&-%)binomialG6$F(%\"nGF&F&" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 14 "sum(f,n=0. .0);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"%\"xGF&F&F%F& " }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 44 "T :=(-1)^k*binomial(n,k)*4^k/binomial(2*k,k);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"TG**)!\"\"%\"kG\"\"\"-%)binomialG6$%\"nGF(F))\"\"%F (F)-F+6$,$*&\"\"#F)F(F)F)F(F'" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 " > " 0 "" {MPLTEXT 1 224 13 "f:=1/(1-2*n);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"fG*&\"\"\"F&,&F&F&*&\"\"#F&%\"nGF&!\"\"F+" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 28 "WZ := WZMe thod(T,f,n,k,'R');" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%#WZG7$*,)!\" \"%\"kG\"\"\"-%)binomialG6$%\"nGF)F*)\"\"%F)F*-F,6$,$*&\"\"#F*F)F*F*F) F(,&F*F**&F5F*F.F*F(F*,$**F)F*,&F4F*F*F(F*,*F7F*F5F*F)F(*(F5F*F.F*F)F* F*F(,&*,F'F*-F,6$,&F.F*F*F*F)F*F/F*F1F(,&F*F(F7F(F*F*F&F(F*F(" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 2 "R;" }} {PARA 214 "" 1 "" {XPPMATH 20 "6#,$**,&*&\"\"#\"\"\"%\"kGF(F(F(!\"\"F( F)F(,&F(F**&F'F(%\"nGF(F(F*,(F-F(F)F*F(F(F*F*" }{TEXT 225 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 14 "sum(T,k=0..0);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"\"" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 14 "sum(f,n=0..0);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"\"" }{TEXT 225 2 " " }}}{EXCHG {PARA 215 "" 0 "" {TEXT 226 30 "Chapter 3 Hypergeometric terms" }}{PARA 213 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 23 "sum(x^k ,k=0..infinity);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%,&%\" xGF%F%!\"\"F(F(" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 42 "convert(sum(x^k,k=0..infinity),hypergeom);" }} {PARA 214 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%,&%\"xGF%F%!\"\"F(F(" } {TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 26 "f:=z ->hypergeom([1],[],z);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"zG6\"6$%)operatorG%&arrowGF(-%*hypergeomG6%7#\"\"\"7\"9$F(F(F(" } {TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 15 "simp lify(f(z));" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%,&F%!\"\"% \"zGF%F'F'" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 60 "convert(sum((-1)^k*x^(2*k)/(2*k)!,k=0..infinity),hypergeom); " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#-%$cosG6#%\"xG" }{TEXT 225 2 " \+ " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 33 "f:=x->hypergeom([], [1/2],-x^2/4);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6 $%)operatorG%&arrowGF(-%*hypergeomG6%7\"7##\"\"\"\"\"#,$*&#F2\"\"%F2*$ )9$F3F2F2!\"\"F(F(F(" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 15 "simplify(f(x));" }}{PARA 214 "" 1 "" {XPPMATH 20 "6 #-%$cosG6#%\"xG" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 64 "convert(sum((-1)^k*x^(2*k+1)/(2*k+1)!,k=0..infinity ),hypergeom);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"xG" } {TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 35 "f:=x ->x*hypergeom([],[3/2],-x^2/4);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>% \"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"-%*hypergeomG6%7\"7## \"\"$\"\"#,$*&#F.\"\"%F.*$)F-F6F.F.!\"\"F.F(F(F(" }{TEXT 225 2 " " }} }{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 15 "simplify(f(x));" }} {PARA 214 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"xG" }{TEXT 225 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 25 "convert(tg(x),hypergeom) ;" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#-%#tgG6#%\"xG" }{TEXT 225 2 " \+ " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 54 "convert(sum(binomia l(n,k)^3,k=0..infinity),hypergeom);" }}{PARA 214 "" 1 "" {XPPMATH 20 " 6#-%*hypergeomG6%7%,$%\"nG!\"\"F'F'7$\"\"\"F+F)" }{TEXT 225 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 54 "convert(sum(binomial(n,k )^1,k=0..infinity),hypergeom);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#)\" \"#%\"nG" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 28 "convert(GAMMA(z),hypergeom);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#-%&GAMMAG6#%\"zG" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 18 "f := BesselJ(a,z);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"fG-%(BesselJG6$%\"aG%\"zG" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 21 "convert(f,hypergeom);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#**-%&GAMMAG6#,&\"\"\"F(%\"aGF(!\"\")%\"zGF) F()\"\"#F)F*-%*hypergeomG6%7\"7#F',$*&\"\"%F*F,F.F*F(" }{TEXT 225 2 " \+ " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 93 "convert(sum((-1)^k *binomial(r-s-k,k)*binomial(r-2*k,n-k)/(r-n-k-1),k=0..infinity),hyperg eom);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#*(-%)binomialG6$%\"rG%\"nG\" \"\",(F'F)F(!\"\"F)F+F+-%*hypergeomG6%7',(F'F+F(F)F)F),&F'F+F(F),&*&\" \"#F+F'F)F+*&F4F+%\"sGF)F),(#F)F4F)F3F+F5F),$F(F+7&,&F3F+F8F),&F'F+F6F ),(F'F+F(F)F4F),$F3F+F)F)" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 86 "convert(sum(binomial(n,k)*binomial(n+a,k)*bino mial(n-a,n-k),k=0..infinity),hypergeom);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#*&-%)binomialG6$,&%\"nG\"\"\"%\"aG!\"\"F(F)-%*hypergeom G6%7%,&F(F+F*F+,$F(F+F17$F),&F)F)F*F+F+F)" }{TEXT 225 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 26 "f:=hypergeom([a,b],[c],1 );" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"fG-%*hypergeomG6%7$%\"aG%\" bG7#%\"cG\"\"\"" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 71 "simplify(f-GAMMA(c-a-b)*GAMMA(c)/GAMMA(c-a)/GAMMA(c -b));#Gauss'identity" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"!" } {TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 31 "f:=h ypergeom([a,b],[1-a+b],-1);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"fG -%*hypergeomG6%7$%\"aG%\"bG7#,(F*\"\"\"F)!\"\"F-F-F." }{TEXT 225 2 " \+ " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 82 "simplify(f-GAMMA(b/ 2+1)*GAMMA(b-a+1)/GAMMA(b+1)/GAMMA(b/2-a+1));#Kummer's identity" }} {PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"!" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 49 "simplify(GAMMA(z)*GAMMA(1-z));# reflection formula" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"\"\"-% $sinG6#*&F$F%%\"zGF%!\"\"" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 31 "assume(c<0);assume(c::integer);" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 43 "f:=hypergeometric([a,b,c],[d,a+ b+c+1-d],1);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"fG-%/hypergeometr icG6%7%%\"aG%\"bG%#c|irG7$%\"dG,,F)\"\"\"F*F/F+F/F/F/F-!\"\"F/" } {TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 88 "simp lify(f-pochhammer(d-a,-c)*pochhammer(d-b,-c)/pochhammer(d,-c)/pochhamm er(d-a-b,-c));" }}{PARA 216 "" 1 "" {XPPMATH 20 "6#,$*,,&*,-%/hypergeo metricG6%7%%\"aG%\"bG%#c|irG7$%\"dG,,F+\"\"\"F,F1F-F1F1F1F/!\"\"F1F1-% &GAMMAG6#,&F/F1F+F2F1-F46#,&F/F1F,F2F1-F46#,&F-F2F/F1F1-F46#,*F-F2F/F1 F+F2F,F2F1F2**-F46#,(F-F2F/F1F+F2F1-F46#,(F-F2F/F1F,F2F1-F46#F/F1-F46# ,(F/F1F+F2F,F2F1F1F1F3F2F7F2F:F2F=F2F2" }{TEXT 227 2 " " }}}{EXCHG {PARA 217 "" 0 "" {TEXT -1 0 "" }}{PARA 217 "" 0 "" {TEXT 228 27 "Chap ter 4: Gosper algorithm" }}{PARA 213 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 8 "restart:" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 213 " > " 0 "" {MPLTEXT 1 224 291 "Sylvester:=proc(f,g) local dg,df,cof,cog, tot,mat,i,j; df:=degree(f,x);dg:=degree(g,x);tot:=df+dg;mat:=matrix(to t,tot,0);for i to dg do for j from i to df+i do mat[i,j]:=coeff(f,x,df +i-j);od;od;for i to df do for j from i to dg+i do mat[dg+i,j]:=coeff( g,x,dg+i-j);od;od;return evalm(mat);end:" }}}{EXCHG {PARA 213 "> " 0 " " {MPLTEXT 1 224 42 "f:=x->(x-1)*(x-2);g:=x->(x-1)*(x^2-x-1)+1;" }} {PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arro wGF(*&,&9$\"\"\"F/!\"\"F/,&F.F/\"\"#F0F/F(F(F(" }{TEXT 225 2 " " }} {PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,&*&,&9$\"\"\"F0!\"\"F0,(*$)F/\"\"#F0F0F/F1F0F1F0F0F0F0F(F(F(" } {TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 30 "S:=S ylvesterMatrix(f(x),g(x));" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"SG- %'RTABLEG6%\"(O[4&-%'MATRIXG6#7'7'\"\"\"!\"$\"\"#\"\"!F17'F1F.F/F0F17' F1F1F.F/F07'F.!\"#F1F0F17'F1F.F5F1F0%'MatrixG" }{TEXT 225 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 46 "expand(simplify((f(z)*g( x)-f(x)*g(z))/(x-z)));" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,4*&)%\"xG \"\"#\"\"\")%\"zGF'F(F(*(\"\"$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(F2F(*&F2F(F*F(F-" } {TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 27 "B:=B ezoutMatrix(f(x),g(x));" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'R TABLEG6%\"(c4C&-%'MATRIXG6#7%7%\"\"#!\"'\"\"'7%!\"\"F.!\"#7%\"\"\"!\"$ F.%'MatrixG" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 30 "Determinant(S);Determinant(B);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"#" }{TEXT 225 2 " " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"#" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 15 "y:=array(1..5);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"yG-% &arrayG6$;\"\"\"\"\"&7\"" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 45 "eqs:=\{seq(sum(S[i,j]*y[j],j=1..5)=0,i=1..5)\} ;" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%$eqsG<'/,(&%\"yG6#\"\"\"F+*&\" \"$F+&F)6#\"\"#F+!\"\"*&F0F+&F)6#F-F+F+\"\"!/,(F.F+*&F-F+F3F+F1*&F0F+& F)6#\"\"%F+F+F5/,(F3F+*&F-F+F:F+F1*&F0F+&F)6#\"\"&F+F+F5/,(F(F+*&F0F+F .F+F1F:F+F5/,(F.F+F2F1FAF+F5" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 36 "sols:=solve(eqs,\{seq(y[j],j=1..5)\});" }} {PARA 214 "" 1 "" {XPPMATH 20 "6#>%%solsG<'/&%\"yG6#\"\"%&F(6#\"\"&/&F (6#\"\"$F+/&F(6#\"\"#F+/&F(6#\"\"\"F+/F+F+" }{TEXT 225 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 20 "subs(sols,evalm(y));" }} {PARA 214 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7'&%\"yG6#\"\"&F'F'F'F'" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 26 "T:= k->(4*k+1)*k!/(2*k+1)!;" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"TGf*6# %\"kG6\"6$%)operatorG%&arrowGF(*(,&*&\"\"%\"\"\"9$F0F0F0F0F0-%*factori alG6#F1F0-F36#,&*&\"\"#F0F1F0F0F0F0!\"\"F(F(F(" }{TEXT 225 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 31 "with(SumTools[Hypergeome tric]);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7;%+AreSimilarG%8Canonical RepresentationG%/ConjugateRTermG%,DefiniteSumG%/ExtendedGosperG%3Exten dedZeilbergerG%'GosperG%.IndefiniteSumG%,IsHolonomicG%5IsHypergeometri cTermG%;IsProperHypergeometricTermG%.IsZApplicableG%,KoepfGosperG%0Koe pfZeilbergerG%+LowerBoundG%-MinimalZpairG% " 0 "" {MPLTEXT 1 224 15 "Gosp er(T(k),k);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\",&*&F%F &%\"kGF&F&F&F&F&-%*factorialG6#F)F&-F+6#F'!\"\"F/" }{TEXT 225 2 " " } }}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 29 "Z:=k->-2*(2*k+1)*k!/(2 *k+1)!;" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"ZGf*6#%\"kG6\"6$%)oper atorG%&arrowGF(,$**\"\"#\"\"\",&*&F.F/9$F/F/F/F/F/-%*factorialG6#F2F/- F46#F0!\"\"F8F(F(F(" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 27 "simplify(Z(k+1)-Z(k)-T(k));" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"!" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 28 "R:=k->simplify(T(k+1)/T(k));" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"RGf*6#%\"kG6\"6$%)operatorG%&arrowGF(-%)simplifyG6# *&-%\"TG6#,&9$\"\"\"F5F5F5-F16#F4!\"\"F(F(F(" }{TEXT 225 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 5 "R(k);" }}{PARA 214 "" 1 " " {XPPMATH 20 "6#,$**\"\"#!\"\",&*&\"\"%\"\"\"%\"kGF*F*\"\"&F*F*,&*&F% F*F+F*F*\"\"$F*F&,&F(F*F*F*F&F*" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 40 "(z,a,b,c):=PolynomialNormalForm(R(k), k);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>6&%\"zG%\"aG%\"bG%\"cG6&#\"\" \"\"\"%F+,&%\"kGF+#\"\"$\"\"#F+,&F.F+F*F+" }{TEXT 225 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 30 "RationalCanonicalForm(R( k),k);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6'#\"\"\"\"\"%F$,&%\"kGF$#\" \"$\"\"#F$,&F'F$F#F$F$" }{TEXT 225 2 " " }}}{EXCHG {PARA 218 "" 0 "" {TEXT -1 0 "" }}{PARA 218 "" 0 "" {TEXT 257 50 "Bernoulli Polynomials \+ and Summation of Polynomials" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 219 "" 0 "" {TEXT 200 79 "We attack the summation prob lem for hypergeometric terms reduced to polynomials" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 230 31 "with(SumTools[Hypergeometric]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#76%2AccurateSummationG%+AreSimilarG%8C anonicalRepresentationG%,DefiniteSumG%/ExtendedGosperG%'GosperG%.Indef initeSumG%,IsHolonomicG%5IsHypergeometricTermG%;IsProperHypergeometric TermG%.IsZApplicableG% \+ " 0 "" {MPLTEXT 1 0 18 "p:=x->a*x^2+b*x+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&%\"aG\"\" \")9$\"\"#F/F/*&%\"bGF/F1F/F/%\"cGF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "simplify(p(x+1)-p(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\"#\"\"\"%\"aGF&%\"xGF&F&F'F&%\"bGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "t:=n->n^2+n+3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,(*$)9$\"\"#\" \"\"F1F/F1\"\"$F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " Gosper(t(n),n)-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\")\"\"$\" \"\"%\"nGF(F(*&#F(F'F(*$)F)F'F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "z:=n->8/3*n+1/3*n^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,&*&#\"\")\"\"$\"\"\"9$F1F 1*&#F1F0F1*$)F2F0F1F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(z(n+1)-z(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)% \"nG\"\"#\"\"\"F(F&F(\"\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sum(t(k),k=0..n);(z(n+1)-z(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\")\"\"\"\"\"$!\"\"%\"nGF&F&#F%F'F&*&F'F(,&F)F&F&F&F'F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\")\"\"\"\"\"$!\"\"%\"nGF&F&#F%F 'F&*&F'F(,&F)F&F&F&F'F&" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 232 77 "dfp:=proc(x,k) if k=0 then return 1 else return product(x-i,i=0..k -1);fi;end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dfpGf*6$%\"xG%\"kG6 \"F)F)@%/9%\"\"!O\"\"\"O-%(productG6$,&9$F/%\"iG!\"\"/F6;F-,&F,F/F/F7F )F)F)" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 232 77 "afp:=proc(x,k) if k=0 then return 1 else return product(x+i,i=0..k-1);fi;end;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$afpGf*6$%\"xG%\"kG6\"F)F)@%/9%\"\"! O\"\"\"O-%(productG6$,&9$F/%\"iGF//F6;F-,&F,F/F/!\"\"F)F)F)" }}} {EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 232 26 "afp(x,2);expand(dfp(x,4) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\",&F$F%F%F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"%\"\"\"F(*&\"\"'F()F&\" \"$F(!\"\"*&\"#6F()F&\"\"#F(F(*&F*F(F&F(F-" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 232 56 "[seq([seq(coeff(expand(dfp(x,n)),x,k),k=0..6)] ,n=0..6)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7)7)\"\"\"\"\"!F&F&F&F&F &7)F&F%F&F&F&F&F&7)F&!\"\"F%F&F&F&F&7)F&\"\"#!\"$F%F&F&F&7)F&!\"'\"#6F .F%F&F&7)F&\"#C!#]\"#N!#5F%F&7)F&!$?\"\"$u#!$D#\"#&)!#:F%" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 232 15 "with(combinat);" }}{PARA 7 "" 1 "" {TEXT -1 67 "Warning, the protected name Chi has been redefined a nd unprotected\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7C%$ChiG%%bellG%)b inomialG%)cartprodG%*characterG%'chooseG%,compositionG%)conjpartG%+dec odepartG%+encodepartG%*fibonacciG%*firstpartG%)graycodeG%)inttovecG%)l astpartG%,multinomialG%)nextpartG%)numbcombG%)numbcompG%)numbpartG%)nu mbpermG%*partitionG%(permuteG%)powersetG%)prevpartG%)randcombG%)randpa rtG%)randpermG%-setpartitionG%*stirling1G%*stirling2G%(subsetsG%)vecto intG" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 232 41 "seq([seq(stirli ng1(n,k),k=0..6)],n=0..6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6)7)\"\"\" \"\"!F%F%F%F%F%7)F%F$F%F%F%F%F%7)F%!\"\"F$F%F%F%F%7)F%\"\"#!\"$F$F%F%F %7)F%!\"'\"#6F-F$F%F%7)F%\"#C!#]\"#N!#5F$F%7)F%!$?\"\"$u#!$D#\"#&)!#:F $" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 232 25 "A:=matrix(4,4,stir ling1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7&\"\" \"\"\"!F+F+7&!\"\"F*F+F+7&\"\"#!\"$F*F+7&!\"'\"#6F2F*" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 232 18 "bb:=evalm(A^(-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bbG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F*F*F+F+ 7&F*\"\"$F*F+7&F*\"\"(\"\"'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "seq([seq(stirling2(n,k),k=1..4)],n=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7&\"\"\"\"\"!F%F%7&F$F$F%F%7&F$\"\"$F$F%7&F$\"\"(\"\"'F $" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 232 39 "simplify(add(bb[4, j]*dfp(x,j),j=1..4));" }}{PARA 220 "" 1 "" {XPPMATH 20 "6#*$%\"xG\"\"% " }{TEXT 231 1 " " }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 232 58 "se q((k+2)*factor(Gosper(product(n-i,i=0..k),n)-1),k=0..5);" }}{PARA 220 "" 1 "" {XPPMATH 20 "6(*&%\"nG\"\"\",&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%*0F$F%F&F%F)F%F,F%F/F%F2F%,&F$F%!\"'F%F %" }{TEXT 231 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "seq(G osper(n^k,n)-1,k=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%\"nG,&*&# \"\"\"\"\"#F'F#F'!\"\"*&#F'F(F'*$)F#F(F'F'F',(*&#F'\"\"'F'F#F'F'*&#F'F (F'F,F'F)*&#F'\"\"$F'*$)F#F6F'F'F',(*&#F'\"\"%F'F,F'F'*&#F'F(F'F7F'F)* &F;F'*$)F#F " 0 "" {MPLTEXT 1 224 36 "seq(factor(Gosper(n^k,n)-1),k=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%\"nG,$*(\"\"#!\"\"F#\"\"\",&F#F(F(F'F(F(,$**\"\"'F'F#F(,&*&F&F(F#F (F(F(F'F(F)F(F(,$*(\"\"%F'F#F&F)F&F(,$*,\"#IF'F#F(F-F(F)F(,(*&\"\"$F() F#F&F(F(*&F7F(F#F(F'F(F'F(F(" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 58 "seq(factor(bernoulli(k+1,n)-bernoulli(k+1))/(k+1),k=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%\"nG,$*(\"\"#!\"\"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(,$* ,\"#IF'F#F(F-F(F)F(,(*&\"\"$F()F#F&F(F(*&F7F(F#F(F'F(F'F(F(" }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 38 "seq(factor(Gosper(2^n*n^ k,n)),k=1..4);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6&*&,&\"\"#!\"\"%\"nG \"\"\"F()F%F'F(*&,(\"\"'F(*&\"\"%F(F'F(F&*$)F'F%F(F(F(F)F(*&,*\"#EF&*& \"#=F(F'F(F(*&F,F(F0F(F&*$)F'\"\"$F(F(F(F)F(*&,,\"$]\"F(*&\"$/\"F(F'F( F&*&\"#OF(F0F(F(*&\"\")F(F8F(F&*$)F'F.F(F(F(F)F(" }{TEXT 225 2 " " }} }{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 30 "Gosper(1/(n^2+sqrt(5)*n +1),n);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,$**\"\"%!\"\",(*&\"\"#\" \"\"%\"nGF*F*F*F**$\"\"&#F*F)F*F*,(\"\"$F&F,F*F(F*F*,(*$)F+F)F*F**&F-F .F+F*F*F*F*F&F*" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 61 "Z:=n->1/4*(2*n+1+5^(1/2))*(-3+5^(1/2)+2*n)/(n^2+5^( 1/2)*n+1);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"ZGf*6#%\"nG6\"6$%)o peratorG%&arrowGF(,$*&#\"\"\"\"\"%F/*(,(*&\"\"#F/9$F/F/F/F/*$\"\"&#F/F 4F/F/,(\"\"$!\"\"F6F/F3F/F/,(*$)F5F4F/F/*&F7F8F5F/F/F/F/F;F/F/F(F(F(" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 38 "sim plify(limit(Z(m),m=infinity)-Z(0));" }}{PARA 214 "" 1 "" {XPPMATH 20 " 6#,&*&\"\"#!\"\"\"\"&#\"\"\"F%F)F(F)" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 69 "evalf(seq(add(1/(n^2+sqrt(5)*n+ 1),n=1..m)-1/2*5^(1/2)+1/2,m=1..100));" }}{PARA 216 "" 1 "" {XPPMATH 20 "6`q$!+0,m>Q!#5$!+:?$Rw#F%$!+ROUl@F%$!+/@)*z3tg)F%$!*#[;DzF%$!*!Q?VtF%$!*7l3%oF %$!*ZaGS'F%$!*#*ev,'F%$!*9-gn&F%$!*KO6P&F%$!*m]t4&F%$!*)G7][F%$!*Zndi% F%$!*p]7U%F%$!*g_SB%F%$!+AF1iS!#6$!+#4+N!RFR$!+*Q^ov$FR$!+kC#3i$FR$!+a +I%\\$FR$!+a7KwLFR$!+W*[gE$FR$!+9>viJFR$!+C')ylIFR$!+kQfuHFR$!+Fxm))GF R$!+dkc2GFR$!+oZ*3t#FR$!+3%*HeEFR$!+3PY*e#FR$!+yH5CDFR$!+e0'>Y#FR$!+eW !GS#FR$!+oWUYBFR$!+Q'HEH#FR$!+ehCTAFR$!+[a6#>#FR$!+QD4X@FR$!+QY/+@FR$! +q)\\o0#FR$!+$>'R:?FR$!+=/ev>FR$!+^tIP>FR$!+J!*[+>FR$!+rS/l=FR$!+Jq*3$ =FR$!+,z(zz\"FR$!+@;AmF\"FR$!+Zn*f D\"FR$!+ZrTS7FR$!+9$>_A\"FR$!+%Q*Q57FR$!+kT\"f>\"FR$!+u5y\"=\"FR$!+F\" yz;\"FR$!+()Q\\a6FR$!+WuJT6FR$!+y$Q%G6FR$!+Gn%e6\"FR$!+))H`.6FR$!+e!)[ \"4\"FR$!+CKqz5FR$!+a,MQ+\"FR$!+M^dQ**!#7$!+M=xS)*Few" }{TEXT 227 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 16 "Gosper(n^2+1,n);" }} {PARA 214 "" 1 "" {XPPMATH 20 "6#,*\"\"\"F$*&#\"\"(\"\"'F$%\"nGF$F$*&# F$\"\"#F$*$)F)F,F$F$!\"\"*&#F$\"\"$F$*$)F)F2F$F$F$" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 20 "Gosper(1/(n^2+1),n); " }}{PARA 222 "" 1 "" {TEXT 234 37 "Error, (in Gosper) no solution fou nd\n" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 22 "Gosper(1/(n^2-n -2),n);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,$*,,&%\"nG\"\"\"F'F'F',*! \"#F'F&\"#7*$F&\"\"#!#7*$F&\"\"$F/F'F&!\"\",&F&F'F0F'F0,(F+F'F&F0F)F'F 0#F'F/" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 32 "Gosper((n-r/2)*binomial(r,n),r);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#*0,,*&\"\"#\"\"\")%\"nGF&F'!\"\"*&\"\"%F'F)F'F**&%\"rGF 'F)F'F'F.F'F'F*F',&F.F'F)F*F',&*&F&F'F)F'F*F.F'F*,&F)F'F&F'F*,&F)F'F'F 'F*,&F)F'*&F&F*F.F'F*F'-%)binomialG6$F.F)F'" }{TEXT 225 2 " " }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 32 "Gosper((n-r/2)*binomial( r,n),n);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#**%\"nG\"\"\",&*&\"\"#F%F $F%!\"\"%\"rGF%F),&F$F%*&F(F)F*F%F)F%-%)binomialG6$F*F$F%" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 31 "a:=64; Gosper( (n+1)/(n^2-a),n);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#>%\"aG\"#k" } {TEXT 225 2 " " }}{PARA 222 "" 1 "" {TEXT 234 63 "Error, (in SumTools :-Hypergeometric:-Gosper) no solution found\n" }}}{EXCHG {PARA 213 "> \+ " 0 "" {MPLTEXT 1 224 44 "Gosper((6*n+3)/(4*n^4+8*n^3+8*n^2+4*n+3),n); " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#*,,(\"\"$\"\"\"*&\"\"#F&)%\"nGF(F &F&*&\"\"%F&F*F&F&F&,&F&F&*&F%F&F)F&F&F&,&*&F(F&F*F&F&F&F&!\"\",&*&\" \"'F&F*F&F&F%F&F&,,*&F,F&)F*F,F&F&*&\"\")F&)F*F%F&F&*&F9F&F)F&F&F+F&F% F&F1" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 58 "seq(simplify(bernoulli(k+1,n+1)-bernoulli(k+1,n)),k=0..5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(\"\"\",$*&\"\"#F#%\"nGF#F#,$*&\"\"$F#) F'F&F#F#,$*&\"\"%F#)F'F*F#F#,$*&\"\"&F#)F'F.F#F#,$*&\"\"'F#)F'F2F#F#" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 235 34 "Per cui Delta_n B(k+1,n)/(k+ 1)=n^k" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 20 "5*(Gosper(n^4 ,n)-1);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"'F&%\"nGF&! \"\"*&#\"\"&\"\"$F&*$)F(F-F&F&F&*$)F(F,F&F&*&#F,\"\"#F&*$)F(\"\"%F&F&F )" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 11 " with(gfun);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#7P%(LaplaceG%.algebrai csubsG%.algeqtodiffeqG%.algeqtoseriesG%.algfuntoalgeqG%&borelG%.cauchy productG%.diffeq*diffeqG%.diffeq+diffeqG%,diffeqtableG%2diffeqtohomdif feqG%,diffeqtorecG%)guesseqnG%(guessgfG%0hadamardproductG%0holexprtodi ffeqG%)invborelG%,listtoalgeqG%-listtodiffeqG%0listtohypergeomG%+listt olistG%.listtoratpolyG%*listtorecG%-listtoseriesG%,maxdegcoeffG%*maxde geqnG%,maxordereqnG%,mindegcoeffG%*mindegeqnG%,minordereqnG%*optionsgf G%,poltodiffeqG%)poltorecG%/ratpolytocoeffG%(rec*recG%(rec+recG%,recto diffeqG%,rectohomrecG%*rectoprocG%.seriestoalgeqG%/seriestodiffeqG%2se riestohypergeomG%-seriestolistG%0seriestoratpolyG%,seriestorecG%/serie stoseriesG" }{TEXT 225 2 " " }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 224 37 "series((t*exp(n*t))/(exp(t)-1),t,10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+7%\"tG\"\"\"\"\"!,&%\"nGF%#F%\"\"#!\"\"F%,(#F%\"#7F%*& #F%F*F%*$)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;,*#F%\"$?(F+*&#F%\"#CF%*$)F(F?F%F%F%*&FDF %F1F%F%*&#F%F.F%F9F%F+F?,**&#F%\"$?\"F%*$)F(\"\"&F%F%F%*&#F%FBF%F(F%F+ *&#F%\"#sF%F9F%F%*&#F%\"#[F%FFF%F+FQ,,#F%\"&S-$F%*&#F%\"$)GF%FFF%F%*&# F%\"%S9F%F1F%F+*&#F%FBF%*$)F(F8F%F%F%*&#F%\"$S#F%FOF%F+F8,,*&#F%\"%?VF %F9F%F+*&FenF%F(F%F%*&#F%\"%S]F%*$)F(\"\"(F%F%F%*&#F%F\\oF%FOF%F%*&#F% F\\oF%F_oF%F+F^p,.#F%\"(+'47F+*&#F%\"&!G " 0 "" {MPLTEXT 1 224 38 "seq (sort(bernoulli(k,n)/k!,n),k=0..8);" }}{PARA 214 "" 1 "" {XPPMATH 20 " 6+\"\"\",&%\"nGF##!\"\"\"\"#F#,(*$F%F(#F#F(F%F&#F#\"#7F#,(*$F%\"\"$#F# \"\"'F*#F'\"\"%F%F,,**$F%F4#F#\"#CF/#F'F-F*F7#F'\"$?(F#,**$F%\"\"&#F# \"$?\"F6#F'\"#[F/#F#\"#sF%F:,,*$F%F2#F#F;F=#F'\"$S#F6#F#\"$)GF*#F'\"%S 9#F#\"&S-$F#,,*$F%\"\"(#F#\"%S]FFFLF=#F#FMF/#F'\"%?VF%FN,.*$F%\"\")#F# \"&?.%FQ#F'\"&!35FF#F#\"%S')F6#F'\"&!G " 0 "" {MPLTEXT 1 224 21 "Gosp er(t*exp(x*t),x);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#*(%\"tG\"\"\"-%$ expG6#*&%\"xGF%F$F%F%,&-F'6#F$F%!\"\"F%F." }{TEXT 225 2 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "simplify(t/(exp(t)-1)-t*exp( t)/(exp(t)-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"tG!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(gamma);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+\\m:sd!#5" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 256 34 "Similarity of hypergeometric terms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "with(SumToo ls[Hypergeometric]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#76%2AccurateSu mmationG%+AreSimilarG%8CanonicalRepresentationG%,DefiniteSumG%/Extende dGosperG%'GosperG%.IndefiniteSumG%,IsHolonomicG%5IsHypergeometricTermG %;IsProperHypergeometricTermG%.IsZApplicableG% " 0 "" {MPLTEXT 1 0 11 "t:=n->n!+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,&- %*factorialG6#9$\"\"\"F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "s:=binomial(n,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"sG-%)binomialG6$%\"nG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "AreSimilar(s,1,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "IsHypergeometricTerm(t( k),k,'c');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "IsHypergeometricTerm(s,n,'c');" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"nG\"\"\"F &F&F&,&F%F&F&!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "t:= k->(-1)^(k-1)*bernoulli(2*k)*4^k*(4^k-1)/(2*k)!*x^(2*k-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tGf*6#%\"kG6\"6$%)operatorG%&arrowGF(*.) !\"\",&9$\"\"\"F1F.F1-%*bernoulliG6#,$*&\"\"#F1F0F1F1F1)\"\"%F0F1,&F8F 1F1F.F1-%*factorialGF4F.)%\"xG,&*&F7F1F0F1F1F1F.F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "printlevel:=1:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 33 "IsHypergeometricTerm(t(k),k,'c');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(t(k),k=0..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 -\"\"\",$*&\"\"#!\"\"%\"xGF&F',$*&\"#CF'F(\"\"%F#,$*&\"$?(F'F(\"\"'F', $*&\"&?.%F'F(\"\")F#,$*&\"(+)GOF'F(\"#5F',$*&\"*+;+z%F'F(\"#7F#,$*&\", +7Hyr)F'F(\"#9F',$*&\"/+!)))*yA4#F'F(\"#;F#,$*&\"1+!GdqtBS'F'F(\"#=F', $*&\"4++kw\"3?!HV#F'F(\"#?F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "t:=k->(-1)^k*x^(2*k)/(2*k)!;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"tGf*6#%\"kG6\"6$%)operatorG%&arrowGF(*()!\"\"9$\"\"\")%\"xG,$*&\" \"#F0F/F0F0F0-%*factorialG6#F3F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "convert(sum(t(k),k=0..infinity),hypergeom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$cosG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "IsHypergeometricTerm(t(k),k,'cer');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "cer;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#!\"\"%\"xGF%,&%\"kG \"\"\"F*F*F&,&*&F%F*F)F*F*F*F*F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(t(k+1)/t(k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#!\"\"%\"xGF%,&%\"kG\"\"\"F*F*F&,&*&F%F*F)F*F*F*F*F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "cer;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#!\"\"%\"xGF%,&%\"kG\"\"\"F*F*F&,&*&F%F*F)F*F*F *F*F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "t:=n!;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG-%*factorialG6#%\"nG" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 258 20 "Zeilberger Algorithm" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "with(SumTools[Hypergeometric]):\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "T :=(n,k)-> binomial(n,k)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"TGf*6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF)*$)-%)binomialG6$9$9% \"\"#\"\"\"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Zpair \+ := Zeilberger(T(n,k),n,k,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Zpa irG7$,(*&,&%\"nG\"\"\"F*F*F*%\"NGF*F*\"\"#!\"\"*&\"\"%F*F)F*F-**%\"kGF ,,(*&\"\"$F*F)F*F*F4F**&F,F*F1F*F-F*,**&F,F*)-%)binomialG6$F)F1F,F*F-* (F/F*F8F*F)F*F-*&)-F:6$F(F1F,F*F)F*F**$F>F*F*F*,4*&F,F*)F1F,F*F**(F/F* FDF*F)F*F**(\"#7F*F)F*F1F*F-*(\"\")F*F1F*)F)F,F*F-*&F/F*F1F*F-*&\"\"(F *FJF*F**&F4F*)F)F4F*F*F*F**&\"\"&F*F)F*F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Zpair[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,& %\"nG\"\"\"F'F'F'%\"NGF'F'\"\"#!\"\"*&\"\"%F'F&F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "simplify(Zpair[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**%\"kG\"\"#,(*&\"\"$\"\"\"%\"nGF)F)F(F)*&F%F)F$F)!\"\" F),**&F%F))-%)binomialG6$F*F$F%F)F,*(\"\"%F)F/F)F*F)F,*&)-F16$,&F*F)F) F)F$F%F)F*F)F)*$F6F)F)F),4*&F%F))F$F%F)F)*(F4F)F=F)F*F)F)*(\"#7F)F*F)F $F)F,*(\"\")F)F$F))F*F%F)F,*&F4F)F$F)F,*&\"\"(F)FCF)F)*&F(F))F*F(F)F)F )F)*&\"\"&F)F*F)F)F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "G: =(n,k)->-k^2*(-3*n-3+2*k)*(-4*binomial(n,k)^2*n-2*binomial(n,k)^2+bino mial(n+1,k)^2*n+binomial(n+1,k)^2)/(4*k^2*n+2*k^2-8*k*n^2-12*n*k-4*k+3 *n^3+7*n^2+5*n+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GGf*6$%\"nG% \"kG6\"6$%)operatorG%&arrowGF),$**9%\"\"#,(*&\"\"$\"\"\"9$F4!\"\"F3F6* &F0F4F/F4F4F4,**(\"\"%F4)-%)binomialG6$F5F/F0F4F5F4F6*&F0F4F;F4F6*&)-F =6$,&F5F4F4F4F/F0F4F5F4F4*$FAF4F4F4,4*(F:F4)F/F0F4F5F4F4*&F0F4FHF4F4*( \"\")F4F/F4)F5F0F4F6*(\"#7F4F5F4F/F4F6*&F:F4F/F4F6*&F3F4)F5F3F4F4*&\" \"(F4FLF4F4*&\"\"&F4F5F4F4F4F4F6F6F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "rr:=(n,k)->(n+1)*T(n+1,k)-2*(2*n+1)*T(n,k)-G(n,k+1)+G (n,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rrGf*6$%\"nG%\"kG6\"6$%)o peratorG%&arrowGF),**&,&9$\"\"\"F1F1F1-%\"TG6$F/9%F1F1*(\"\"#F1,&*&F7F 1F0F1F1F1F1F1-F36$F0F5F1!\"\"-%\"GG6$F0,&F5F1F1F1F<-F>F;F1F)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "seq(rr(21,k),k=0..11);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6.\"\"!F#F#F#F#F#F#F#F#F#F#F#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Verify(T(n,k),Zpair,n,k,N); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "F:=(n,k)->binomial(x,k)*binomial(y,n-k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6$%\"nG%\"kG6\"6$%)operatorG%&arrowG F)*&-%)binomialG6$%\"xG9%\"\"\"-F/6$%\"yG,&9$F3F2!\"\"F3F)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "IsHypergeometricTerm(F(n,k), n,'c');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(% \"yG\"\"\"%\"nG!\"\"%\"kGF&F&,(F'F&F)F(F&F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "IsHypergeometricTerm(F(n,k),k,'c');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"xG\"\"\"%\"kG!\"\" F&,&F'F&F&F&F(,*%\"yGF&%\"nGF(F'F&F&F&F(,&F,F&F'F(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Zpair:=Zeilberger(F(n,k),n,k,N):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "simplify(sum(F(n,k),k=0..inf inity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%&GAMMAG6#,&%\"nG\"\"\" F)F)!\"\"-F%6#,(%\"yGF)%\"xGF)F)F)F)-F%6#,*F/F)F)F)F.F)F(F*F*" }}} {EXCHG {PARA 257 "" 0 "" {TEXT 259 25 "Petkovsek Algorithm Hyper" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "with(SumTools[Hypergeometric]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#76%2AccurateSummationG%+AreSimilarG%8CanonicalR epresentationG%,DefiniteSumG%/ExtendedGosperG%'GosperG%.IndefiniteSumG %,IsHolonomicG%5IsHypergeometricTermG%;IsProperHypergeometricTermG%.Is ZApplicableG% " 0 "" {MPLTEXT 1 0 58 "F:=(n,k)->binomial(3*k+1,k)*binomial(3*n-3*k,n-k)/(3* k+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6$%\"nG%\"kG6\"6$%)op eratorG%&arrowGF)*(-%)binomialG6$,&*&\"\"$\"\"\"9%F4F4F4F4F5F4-F/6$,&* &F3F49$F4F4*&F3F4F5F4!\"\",&F:F4F5F " 0 "" {MPLTEXT 1 0 32 "Zpair:=Zeilberger(F(n,k),n,k,N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "L:=Zpair[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,.*&,**&\"#;\"\"\")%\"nG\"\"$F*!\"\"*&\"#'*F*)F, \"\"#F*F.*&\"$)=F*F,F*F.\"$?\"F.F*)%\"NGF2F*F**&,**&\"$;#F*F+F*F**&\"$ s*F*F1F*F**&\"%+:F*F,F*F*\"$#zF*F*F7F*F**&\"$H(F*F+F*F.*&\"%(=#F*F1F*F .*&\"%1@F*F,F*F.\"$['F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " seq(factor(coeff(L,N,i)),i=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%, $**\"#\")\"\"\",&*&\"\"$F&%\"nGF&F&\"\"#F&F&,&*&F)F&F*F&F&\"\"%F&F&,&F *F&F&F&F&!\"\",$*(\"#7F&,&*&F+F&F*F&F&F)F&F&,(*&\"\"*F&)F*F+F&F&*&\"#F F&F*F&F&\"#AF&F&F&,$**F.F&,&*&F+F&F*F&F&\"\"&F&F&F4F&,&F*F&F+F&F&F0" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(LREtools);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#75%*REcontentG%)REcreateG%'REplotG%+REprimpa rtG%.REreduceorderG%'REtoDEG%*REtodeltaG%)REtoprocG%/autodispersionG%. constcoeffsolG%&deltaG%+dispersionG%(divconqG%)firstlinG%.hypergeomsol sG%)polysolsG%,ratpolysolsG%(riccatiG%&shiftG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "rec:=(-16*n^3-96*n^2-188*n-120)*f(n+2)+(216*n^3 +972*n^2+1500*n+792)*f(n+1)+(-729*n^3-2187*n^2-2106*n-648)*f(n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$recG,(*&,**&\"#;\"\"\")%\"nG\"\"$F* !\"\"*&\"#'*F*)F,\"\"#F*F.*&\"$)=F*F,F*F.\"$?\"F.F*-%\"fG6#,&F,F*F2F*F *F**&,**&\"$;#F*F+F*F**&\"$s*F*F1F*F**&\"%+:F*F,F*F*\"$#zF*F*-F76#,&F, F*F*F*F*F**&,**&\"$H(F*F+F*F.*&\"%(=#F*F1F*F.*&\"%1@F*F,F*F.\"$['F.F*- F76#F,F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "hh:=hypergeom sols(rec,f(n),\{\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#hhG,&*&#\" \"\"\"\"#F(*,,&*&\"\")F(-%\"fG6#\"\"!F(!\"\"*&F)F(-F/6#F(F(F(F()#\"#F \"\"%%\"nGF(-%&GAMMAG6#,&F:F(F(F(F(-F<6#,&F:F(#\"\"$F)F(F2%#PiGF'F(F(* &#FCF9F(*8,&*&F)F(F4F(F2*&\"\"*F(F.F(F(F(F6F()#F)FKF:F()FCF:F)-F<6#,&F :F(#F9FCF(F(-F<6#,&F:F(#F)FCF(F()F)F:F2F?F2F;F2FD#F2F)FCF'F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "hh[1];hh[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*()#\"#F\"\"%%\"nG\"\"\"-%&GAMMAG6#,&F(F)F)F)F)-F+ 6#,&F(F)#\"\"$\"\"#F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,)#\"#F \"\"%%\"nG\"\"\"-%&GAMMAG6#,&F(F)#F'\"\"$F)F)-F+6#,&F(F)#\"\"#F/F)F)-F +6#,&F(F)F)F)!\"\"-F+6#,&F(F)#F/F4F)F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "h1:=n->(27/4)^n*GAMMA(n+1)/GAMMA(n+3/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h1Gf*6#%\"nG6\"6$%)operatorG%&arrowGF(*()#\" #F\"\"%9$\"\"\"-%&GAMMAG6#,&F1F2F2F2F2-F46#,&F1F2#\"\"$\"\"#F2!\"\"F(F (F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "h2:=n->(27/4)^n*GAMM A(n+4/3)*GAMMA(n+2/3)/GAMMA(n+1)/GAMMA(n+3/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h2Gf*6#%\"nG6\"6$%)operatorG%&arrowGF(*,)#\"#F\"\"%9 $\"\"\"-%&GAMMAG6#,&F1F2#F0\"\"$F2F2-F46#,&F1F2#\"\"#F8F2F2-F46#,&F1F2 F2F2!\"\"-F46#,&F1F2#F8F=F2FAF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s:=n->sum(F(n,k),k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%$sumG6$-%\"F G6$9$%\"kG/F3;\"\"!F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "s(0);s(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "simplify(subs(\{f(0)=1,f(1)=4\},1/2*(-8*f(0)+2*f(1))*(27/4)^n*GAM MA(n+1)/GAMMA(n+3/2)*Pi^(1/2)+3/4*(-2*f(1)+9*f(0))*(27/4)^n*(2/9)^n*(3 ^n)^2*GAMMA(n+4/3)*GAMMA(n+2/3)/(2^n)/GAMMA(n+3/2)/GAMMA(n+1)/Pi^(1/2) *3^(1/2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"$\"\"#\"\"\"*. F&#F(F'%#PiG!\"\"-%&GAMMAG6#,&*&F'F(%\"nGF(F(F'F(F,-F.6#,&F2F(#\"\"%F& F(F(-F.6#,&F2F(#F'F&F(F()\"#FF2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "ms:=n->3/2*3^(1/2)/Pi/GAMMA(2*n+2)*GAMMA(n+4/3)*GAMMA (n+2/3)*27^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#msGf*6#%\"nG6\"6$% )operatorG%&arrowGF(,$*&#\"\"$\"\"#\"\"\"*.F/#F1F0%#PiG!\"\"-%&GAMMAG6 #,&*&F0F19$F1F1F0F1F5-F76#,&F;F1#\"\"%F/F1F1-F76#,&F;F1#F0F/F1F1)\"#FF ;F1F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(ms(n), n=0..6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)\"\"\"\"\"%\"#@\"$?\"\"$:( \"%oV\"&Kr#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "convert(ms(n ),binomial) assuming n::posint;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* &#\"\"$\"\"#\"\"\"*,%#PiG!\"\"F&#F(F')\"#F%\"nGF(,&F/F(#F'F&F(F+-%)bin omialG6$,&*&F'F(F/F(F(F(F(,&F/F(#F(F&F(F+F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "so ls:=solve(\{a*h1(0)+b*h2(0)=s(0),a*h1(1)+b*h2(1)=s(1)\},\{a,b\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG<$/%\"aG\"\"!/%\"bG,$**\"\"$\" \"\"\"\"%!\"\"%#PiG#F0\"\"#F-#F.F3F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "simplify(subs(sols,a*h1(n)+b*h2(n)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"$\"\"#\"\"\"*.%#PiG!\"\"F&#F(F'-%&GAMMAG6 #,&*&F'F(%\"nGF(F(F'F(F+-F.6#,&F2F(#\"\"%F&F(F(-F.6#,&F2F(#F'F&F(F()\" #FF2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "ss:=n->3/2*1/ Pi*3^(1/2)/GAMMA(2*n+2)*GAMMA(n+4/3)*GAMMA(n+2/3)*27^n;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#ssGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,$*&#\" \"$\"\"#\"\"\"*.%#PiG!\"\"F/#F1F0-%&GAMMAG6#,&*&F0F19$F1F1F0F1F4-F76#, &F;F1#\"\"%F/F1F1-F76#,&F;F1#F0F/F1F1)\"#FF;F1F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "GAMMA(1/3)*GAMMA(2/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\"\"\"$!\"\"%#PiGF&F'#F&F%F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "convert(ss(n),binomial);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"$\"\"#\"\"\"*,%#PiG!\"\"F&#F( F')\"#F%#n|irGF(,&F/F(#F'F&F(F+-%)binomialG6$,&*&F'F(F/F(F(F(F(,&F/F(# F(F&F(F+F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "S:=(n,k)->F (n,k)+F(n,n-k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SGf*6$%\"nG%\"k G6\"6$%)operatorG%&arrowGF),&-%\"FG6$9$9%\"\"\"-F/6$F1,&F1F3F2!\"\"F3F )F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Sympair:=Zeilberge r(S(n,k),n,k,N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Sympair [1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&,(*&\"\"%\"\"\")%\"nG\"\"# F(F(\"\"'F(*&\"#5F(F*F(F(F(%\"NGF(F(*&\"#FF(F)F(!\"\"*&\"#aF(F*F(F2\"# CF2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "res:=(6+4*n^2+10*n)* f(n+1)+(-27*n^2-54*n-24)*f(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$r esG,&*&,(*&\"\"%\"\"\")%\"nG\"\"#F*F*\"\"'F**&\"#5F*F,F*F*F*-%\"fG6#,& F,F*F*F*F*F**&,(*&\"#FF*F+F*!\"\"*&\"#aF*F,F*F9\"#CF9F*-F26#F,F*F*" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "hypergeomsols(res,f(n),\{\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"$\"\"%\"\"\"*6-%\"fG6# \"\"!F()#F&\"\"#%\"nGF()F&F1F0-%&GAMMAG6#,&F1F(#F'F&F(F(-F46#,&F1F(#F0 F&F(F()F0F1!\"\"-F46#,&F1F(F/F(F=-F46#,&F1F(F(F(F=%#PiG#F=F0F&#F(F0F(F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "subs(f(0)=2,3/4*f(0)* (3/2)^n*(3^n)^2*GAMMA(n+4/3)*GAMMA(n+2/3)/(2^n)/GAMMA(n+3/2)/GAMMA(n+1 )/Pi^(1/2)*3^(1/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"$\"\" #\"\"\"*4)F%%\"nGF()F&F+F'-%&GAMMAG6#,&F+F(#\"\"%F&F(F(-F.6#,&F+F(#F'F &F(F()F'F+!\"\"-F.6#,&F+F(F%F(F8-F.6#,&F+F(F(F(F8%#PiG#F8F'F&#F(F'F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "simplify(3/2*(3/2)^n*( 3^n)^2*GAMMA(n+4/3)*GAMMA(n+2/3)/(2^n)/GAMMA(n+3/2)/GAMMA(n+1)/Pi^(1/2 )*3^(1/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*0\"\"$\"\"\")\"#F%\" nGF&F%#F&\"\"#%#PiG!\"\"-%&GAMMAG6#,&*&F+F&F)F&F&F+F&F--F/6#,&F)F&#\" \"%F%F&F&-F/6#,&F)F&#F+F%F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "hh:=n->(27/4)^n*GAMMA(4/3+n)*GAMMA(2/3+n)/GAMMA(n+1)/GAMMA(n+3 /2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#hhGf*6#%\"nG6\"6$%)operator G%&arrowGF(*,)#\"#F\"\"%9$\"\"\"-%&GAMMAG6#,&#F0\"\"$F2F1F2F2-F46#,&# \"\"#F8F2F1F2F2-F46#,&F1F2F2F2!\"\"-F46#,&F1F2#F8F=F2FAF(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "F:=(n,k)->(-1)^k*binomial(n, k)/binomial(x+k,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6$%\"nG %\"kG6\"6$%)operatorG%&arrowGF)*()!\"\"9%\"\"\"-%)binomialG6$9$F0F1-F3 6$,&%\"xGF1F0F1F0F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Zeilberger(F(n,k),n,k,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$!\" \"*,,&%\"xG\"\"\"%\"kGF(F()F$F)F(-%)binomialG6$%\"nGF)F(,&F'F(F.F(F$-F ,6$F&F)F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "G:=(n,k)->(x+k )*(-1)^k*binomial(n,k)/(x+n)/binomial(x+k,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GGf*6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF)*,,&%\"xG \"\"\"9%F0F0)!\"\"F1F0-%)binomialG6$9$F1F0,&F/F0F7F0F3-F56$F.F1F3F)F)F )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "G(n,k)/F(n,k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"%\"kGF&F&,&F%F&%\"nGF&! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "n:=4;x:=sqrt(3);seq (simplify(expand(-F(n,k)-G(n,k+1)+G(n,k))),k=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"x G*$\"\"$#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"!F#F#F#F#F #F#F#F#F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "simplify(expan d(-F(n,k)-G(n,k+1)+G(n,k)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "hh(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"%\"\"\"\"\"*!\"\"%#PiG#F&\"\"#\"\"$F*F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "N:=proc(f,n) return f(n+1);e nd;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NGf*6$%\"fG%\"nG6\"F)F)O-9$ 6#,&9%\"\"\"F0F0F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "L :=(n-5)*(n+1)*N+(n-5)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,&*( ,&%\"nG\"\"\"\"\"&!\"\"F),&F(F)F)F)F)%\"NGF)F)*$)F'\"\"#F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "rr:=(n-5)*(n+1)*f(n+1)+(n-5) ^2*f(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rrG,&*(,&%\"nG\"\"\"\" \"&!\"\"F),&F(F)F)F)F)-%\"fG6#F,F)F)*&)F'\"\"#F)-F.6#F(F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "hypergeomsols(rr,f(n),\{\},o utput=basis);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#*0)!\"\"%\"nG\"\"\" F'F&,&F'F(F(F&F&,&F'F(\"\"#F&F&,&F'F(\"\"$F&F&,&F'F(\"\"%F&F&,&F'F(\" \"&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "rr:=(n-m)*(n+1)* f(n+1)+(n-m)^2*f(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rrG,&*(,&% \"nG\"\"\"%\"mG!\"\"F),&F(F)F)F)F)-%\"fG6#F,F)F)*&)F'\"\"#F)-F.6#F(F)F )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "hypergeomsols(rr,f(n), \{\},output=basis);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#*()!\"\"%\"nG \"\"\"-%&GAMMAG6#,&F'F(%\"mGF&F(-F*6#,&F'F(F(F(F&" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 37 "series((1+x-sqrt(1-6*x+x^2))/4,x,12);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"xG\"\"\"F%F%\"\"#\"\"$F'\"#6\"\"% \"#X\"\"&\"$(>\"\"'\"$.*\"\"(\"%zU\"\")\"&$z?\"\"*\"'\\I5\"#5\"'f)=&F( -%\"OG6#F%\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#77%3HypergeometricTermG%3IsDesingularizableG %*REcontentG%)REcreateG%'REplotG%+REprimpartG%.REreduceorderG%'REtoDEG %*REtodeltaG%)REtoprocG%/autodispersionG%.constcoeffsolG%&deltaG%+disp ersionG%(divconqG%)firstlinG%.hypergeomsolsG%)polysolsG%,ratpolysolsG% (riccatiG%&shiftG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "rec:=( n+2)*a(n+2)-3*(2*n+1)*a(n+1)+(n-1)*a(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$recG,(*&,&%\"nG\"\"\"\"\"#F)F)-%\"aG6#F'F)F)*(\"\"$F),&*&F*F) F(F)F)F)F)F)-F,6#,&F(F)F)F)F)!\"\"*&,&F(F)F)F5F)-F,6#F(F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "hypergeomsols(rec,a(n),\{a(0 )=1,a(1)=1\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%.hypergeomsolsG6%, (*&,&%\"nG\"\"\"\"\"#F*F*-%\"aG6#F(F*F**(\"\"$F*,&*&F+F*F)F*F*F*F*F*-F -6#,&F)F*F*F*F*!\"\"*&,&F)F*F*F6F*-F-6#F)F*F*F9<$/-F-6#\"\"!F*/-F-6#F* F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ode1:=REtoDE(rec=0, a (n), \{\}, f(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode1G-%&DESolG 6$<#,(%\"fG\"\"\"*&,&\"\"$F+%\"zG!\"\"F+-%\"DG6#F*F+F+*&,(*$)F/\"\"#F+ F0*&\"\"'F+F/F+F+F+F0F+--%#@@G6$F2F8F3F+F+<#F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 "rec1 := REcreate(a(n+2)-2*a(n+1)+a(n)=0, a(n), \{\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rec1G-%)REcreateG6%/,(-% \"aG6#,&%\"nG\"\"\"\"\"#F/F/*&F0F/-F+6#,&F.F/F/F/F/!\"\"-F+6#F.F/\"\"! F6<\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "dsolve(f(z)+(3-z)* diff(f(z),z)+(-z^2+6*z-1)*diff(f(z),z,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"zG*&,&**\"#;!\"\"%$_C1G\"\"\",&*&\"\"#F.F'F .F.\"\"'F,F.,(*$)F'F1F.F.*&F2F.F'F.F,F.F.#F,F1F,%$_C2GF.F.F3#F.F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "an:=z->(-1/16*_C1*(2*z-6)/(z ^2-6*z+1)^(1/2)+_C2)*(z^2-6*z+1)^(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#anGf*6#%\"zG6\"6$%)operatorG%&arrowGF(*&,&*&#\"\"\"\"#;F0*(%$ _C1GF0,&*&\"\"#F09$F0F0\"\"'!\"\"F0,(*$)F7F6F0F0*&F8F0F7F0F9F0F0#F9F6F 0F9%$_C2GF0F0F:#F0F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sols:=solve(\{an(0)=0,subs(z=0,diff(an(z),z))=1\},\{_C1,_C2\});" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG<$/%$_C2G#!\"$\"\")/%$_C1G\" \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "simplify(subs(sols, an(z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\")!\"\"%\"zG\"\"\"F &#\"\"$F%F(*(F*F(F%F&,(*$)F'\"\"#F(F(*&\"\"'F(F'F(F&F(F(#F(F/F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "series(-1/2*z+3/2-1/2*(z^2-6 *z+1)^(1/2),z,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"\"\" \"!F%F%\"\"#F'\"\"'\"\"$\"#A\"\"%\"#!*\"\"&\"$%RF(\"%1=\"\"(\"%e&)\"\" )\"&'eT\"\"*-%\"OG6#F%\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "simplify((-1/16*4*(2*z-6)/(z^2-6*z+1)^(1/2)-3/8*4+1)*(z^2-6*z+1)^( 1/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"\"$\"\"#\"\"\"*&F&!\"\" %\"zGF'F)*&F&F),(*$)F*F&F'F'*&\"\"'F'F*F'F)F'F'#F'F&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rec:=i(n+2)-i(n+1)-(n+1)*i(n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$recG,(-%\"iG6#,&%\"nG\"\"\"\"\"#F+F +-F'6#,&F*F+F+F+!\"\"*&F/F+-F'6#F*F+F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "hh:=hypergeomsols(rec,i(n),\{\});" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#hhG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 223 "" 0 "" {TEXT -1 0 "" }}{PARA 224 "" 0 "" {TEXT -1 0 "" }}}{MARK "236 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }