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One of the most famous su ccesses of this so-called ``WZ-method'' has been to provide a computer proof of the combinatorial identity" }}}{EXCHG {PARA 256 "" 0 "" {XPPEDIT 18 0 "sum(binomial(n,k)^2*binomial(n+k,k)^2,k=0..n)=sum(binom ial(n,k)*binomial(n+k,k)*sum(binomial(k,j)^3,j=0..k),k=0..n)" "/-%$sum G6$*&-%)binomialG6$%\"nG%\"kG\"\"#-F(6$,&F*\"\"\"F+F0F+\"\"#/F+;\"\"!F *-F$6$*(-F(6$F*F+F0-F(6$,&F*F0F+F0F+F0-F$6$*$-F(6$F+%\"jG\"\"$/FB;F4F+ F0/F+;F4F*" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and to prove that the sequence of these numbers " }{XPPEDIT 18 0 "a[n]" " &%\"aG6#%\"nG" }{TEXT -1 47 " satisfies the second order recurrence eq uation" }}}{EXCHG {PARA 257 "" 0 "" {XPPEDIT 18 0 "(n+2)^3*u[n+2]-((n+ 2)^3+(n+1)^3+4*(2*n+3)^3)*u[n+1]+(n+1)^3*u[n]=0" "/,(*&,&%\"nG\"\"\"\" \"#F'\"\"$&%\"uG6#,&F&F'\"\"#F'F'F'*&,(*$,&F&F'\"\"#F'\"\"$F'*$,&F&F' \"\"\"F'\"\"$F'*&\"\"%F'*$,&*&\"\"#F'F&F'F'\"\"$F'\"\"$F'F'F'&F+6#,&F& F'\"\"\"F'F'!\"\"*&,&F&F'\"\"\"F'\"\"$&F+6#F&F'F'\"\"!" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Proving this recurrence was a crucial step of Ap\351ry's proof for the irrationality of" }}}{EXCHG {PARA 259 "" 0 "" {XPPEDIT 18 0 "Zeta(3)=sum(1/k^3,k=1..infinity)" "/- %%ZetaG6#\"\"$-%$sumG6$*&\"\"\"\"\"\"*$%\"kG\"\"$!\"\"/F.;\"\"\"%)infi nityG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "On the \+ other hand, the identity itself stems from a number-theoretic question raised by Schmidt in (Schmidt, Asmus L. (1990): Generalized Legendre \+ polynomials, " }{TEXT 256 21 "J. reine angew. Math." }{TEXT -1 2 ", " }{TEXT 272 3 "404" }{TEXT -1 10 ":192-202)." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 69 "Several proofs of the identity above, that relates the \+ Ap\351ry numbers " }{XPPEDIT 18 0 "a[n]" "&%\"aG6#%\"nG" }{TEXT -1 22 " to the Franel numbers" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "f[ n]=sum(binomial(n,k)^3,k=0..n)" "/&%\"fG6#%\"nG-%$sumG6$*$-%)binomialG 6$F&%\"kG\"\"$/F.;\"\"!F&" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "were given in (Strehl, Volker (1994): Binomial Identities , Combinatorial and Algorithmic Aspects, " }{TEXT 257 14 "Discrete Mat h." }{TEXT -1 2 ", " }{TEXT 271 3 "136" }{TEXT -1 380 ":309-346). One of them in particular is based on Zeilberger's algorithm for hypergeo metric summation, and yields the recurrence equation above as a by-pro duct. In the following sections, we first recall how Ap\351ry was led to the identity, borrowing from Van der Poorten's report (Van der Poo rten, Alfred (1979): A Proof that Euler missed... Ap\351ry's Proof of \+ the Irrationality of " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" } {TEXT -1 2 ", " }{TEXT 259 19 "Math. Intelligencer" }{TEXT -1 2 ", " } {TEXT 273 1 "1" }{TEXT -1 67 ":195-203); we next give a proof for both results using our package " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" "" } {TEXT -1 54 ", and finally exploit the recurrence equation to beat " } {TEXT 258 5 "Maple" }{TEXT -1 26 " computing many digits of " } {XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" }{TEXT -1 1 "." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Sketch of Ap\351ry's Proof" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Ap\351ry's first remark is that the doubl e sequence" }}}{EXCHG {PARA 262 "" 0 "" {XPPEDIT 18 0 "c[n,k]=sum(1/m^ 3,m=1..n)+sum((-1)^(m+1)/(2*m^3*binomial(n,m)*binomial(n+m,m)),m=1..k) " "/&%\"cG6$%\"nG%\"kG,&-%$sumG6$*&\"\"\"\"\"\"*$%\"mG\"\"$!\"\"/F0;\" \"\"F&F.-F*6$*&),$\"\"\"F2,&F0F.\"\"\"F.F.**\"\"#F.*$F0\"\"$F.-%)binom ialG6$F&F0F.-FC6$,&F&F.F0F.F0F.F2/F0;\"\"\"F'F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "tends to " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$ " }{TEXT -1 14 " uniformly in " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 4 " in " } {XPPEDIT 18 0 "``(1..infinity)" "-%!G6#;\"\"\"%)infinityG" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 87 " tends to infin ity. This stems from the alternating series being uniformly bounded b y " }{XPPEDIT 18 0 "1/n" "*&\"\"\"\"\"\"%\"nG!\"\"" }{TEXT -1 99 ". H owever, the convergence of this series is not strong enough so as to s how the irrationality of " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" }{TEXT -1 78 ". Ap\351ry used summation methods to accelerate the con vergence. Namely, define" }}}{EXCHG {PARA 263 "" 0 "" {XPPEDIT 18 0 " a[n]=sum(binomial(n,k)^2*binomial(n+k,k)^2*c[n,k],k=0..n)" "/&%\"aG6#% \"nG-%$sumG6$*(-%)binomialG6$F&%\"kG\"\"#-F,6$,&F&\"\"\"F.F3F.\"\"#&% \"cG6$F&F.F3/F.;\"\"!F&" }{TEXT -1 11 " and " }{XPPEDIT 18 0 "b[ n]=sum(binomial(n,k)^2*binomial(n+k,k)^2,k=0..n" "/&%\"bG6#%\"nG-%$sum G6$*&-%)binomialG6$F&%\"kG\"\"#-F,6$,&F&\"\"\"F.F3F.\"\"#/F.;\"\"!F&" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "then " } {XPPEDIT 18 0 "a[n]/b[n]" "*&&%\"aG6#%\"nG\"\"\"&%\"bG6#F&!\"\"" } {TEXT -1 15 " also tends to " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\" $" }{TEXT -1 100 ". Here appears the crucial recurrence of Ap\351ry: \+ one remarks that it is satisfied by both sequences " }{XPPEDIT 18 0 "a " "I\"aG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 25 ", with initial conditions" }}}{EXCHG {PARA 264 "" 0 "" {XPPEDIT 18 0 "a[0]=0" "/&%\"aG6#\"\"!F&" }{TEXT -1 5 ", " } {XPPEDIT 18 0 "a[1]=6" "/&%\"aG6#\"\"\"\"\"'" }{TEXT -1 11 ", and \+ " }{XPPEDIT 18 0 "b[0]=1" "/&%\"bG6#\"\"!\"\"\"" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "b[1]=5" "/&%\"bG6#\"\"\"\"\"&" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "By a number-theoretic argument, it follows from this recurrence that" }}}{EXCHG {PARA 265 "" 0 "" {XPPEDIT 18 0 "Zeta(3)-a[n]/b[n]=O(q[n]^(-1+delta)" "/,&-%%ZetaG6#\"\" $\"\"\"*&&%\"aG6#%\"nGF(&%\"bG6#F-!\"\"F1-%\"OG6#)&%\"qG6#F-,&\"\"\"F1 %&deltaGF(" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "with " }}}{EXCHG {PARA 266 "" 0 "" {XPPEDIT 18 0 "q[n]=2*lcm(1,2,``..``,n)^ 3*b[n]" "/&%\"qG6#%\"nG*(\"\"#\"\"\"*$-%$lcmG6&\"\"\"\"\"#;%!GF1F&\"\" $F)&%\"bG6#F&F)" }{TEXT -1 11 " and " }{XPPEDIT 18 0 "delta=(4*l n(1+sqrt(2))-3)/(4*ln(1+sqrt(2))+3)" "/%&deltaG*&,&*&\"\"%\"\"\"-%#lnG 6#,&\"\"\"F(-%%sqrtG6#\"\"#F(F(F(\"\"$!\"\"F(,&*&\"\"%F(-F*6#,&\"\"\"F (-F/6#\"\"#F(F(F(\"\"$F(F3" }{TEXT -1 20 ", which is positive." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "This is sufficient to prove that \+ " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" }{TEXT -1 63 " is irratio nal, and yields an irrationality measure of at least" }}}{EXCHG {PARA 267 "" 0 "" {XPPEDIT 18 0 "1+1/delta=13.417820..``" "/,&\"\"\"\"\"\"*& \"\"\"F%%&deltaG!\"\"F%;$\")?yT8!\"'%!G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Proof of Ap\351ry's Recurrence" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "In this session, we chiefly use the user-oriented package " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(Mgfun);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%,diag_of_sysG%+int_of_sysG%+pol_to_sysG%+sum_of_sysG %(sys*sysG%(sys+sysG" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Recurren ce for the Left-Hand Side" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "We fi rst prove that the Ap\351ry numbers, as defined by the left-hand side " }}}{EXCHG {PARA 260 "" 0 "" {XPPEDIT 18 0 "a[n]=sum(binomial(n,k)^2* binomial(n+k,k)^2,k=0..n)" "/&%\"aG6#%\"nG-%$sumG6$*&-%)binomialG6$F&% \"kG\"\"#-F,6$,&F&\"\"\"F.F3F.\"\"#/F.;\"\"!F&" }{TEXT -1 1 "," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "satisfy the announced recurrence. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The summand" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=binomial(n,k)^2*binomial(n+k,k)^2:" ">% \"fG*&-%)binomialG6$%\"nG%\"kG\"\"#-F&6$,&F(\"\"\"F)F.F)\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "satisfies both following equations :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "h(n+1,k)/h(n,k)=factor (normal(subs(n=n+1,f)/f,expanded));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*&-%\"hG6$,&%\"nG\"\"\"F*F*%\"kGF*-F&6$F)F+!\"\"*&,(F)F*F*F*F+F*\"\" #,(F)F.F.F*F+F*!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "h(n, k+1)/h(n,k)=factor(normal(subs(k=k+1,f)/f,expanded));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"hG6$%\"nG,&%\"kG\"\"\"F+F+F+-F&6$F(F*!\"\"* (,(F(F+F+F+F*F+\"\"#,&F(F.F*F+F1F)!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "This yields the following system" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sys:=collect(map(numer,map(eq->op(1,eq)-op(2,eq) ,\{\"\",\"\})),h);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG<$,&*&,.* $%\"nG\"\"#\"\"\"F*F+*&F*F,%\"kGF,!\"#F,F,F.F/*$F.F+F,F,-%\"hG6$,&F*F, F,F,F.F,F,*&,.F-F/!\"\"F,F)F7F*F/F.F/F0F7F,-F26$F*F.F,F,,&*&,6*$F*\"\" %F7*$F*\"\"$F/F)F7*&F*F+F.F,F+*&F*F+F.F+F+F-F+*&F*F,F.F+F+F0F7*$F.F@F/ *$F.F>F7F,F8F,F,*&,,FEF,FDF>F0\"\"'F.F>F,F,F,-F26$F*,&F.F,F,F,F,F," }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "where each element " }{XPPEDIT 18 0 "expr" "I%exprG6\"" }{TEXT -1 33 " in the set denotes the equatio n " }{XPPEDIT 18 0 "expr=0" "/%%exprG\"\"!" }{TEXT -1 31 ". The defin ite summation over " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "``(0,n)" "-%!G6$\"\"!%\"nG" }{TEXT -1 39 " is performe d by the following call to " }{HYPERLNK 17 "Mgfun[sum_of_sys]" 2 "Mgfu n[sum_of_sys]" "" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sum_of_sys(sys,k=-infinity..infinity,takayama_algo); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#,(*&,**$%\"nG\"\"$\"\"\"*$F(\"\" #F)F(F)F*F*F*-%\"hG6#F(F*F**&,*F'!#MF(!$J#F+!$`\"!$<\"F*F*-F.6#,&F(F*F *F*F*F**&,*F'F*F+\"\"'F(\"#7\"\")F*F*-F.6#,&F(F*F,F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "rec[left]:=op(collect(\",h,factor)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$recG6#%%leftG,(*&,&%\"nG\"\" \"F,F,\"\"$-%\"hG6#F+F,F,*&,&F+F,\"\"#F,F--F/6#F2F,F,*(,&F+F3F-F,F,,(* $F+F3\"# " 0 "" {XPPEDIT 19 1 "f:=binomial(n,k)^3:" ">%\"fG*$ -%)binomialG6$%\"nG%\"kG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "h(n+1,k)/h(n,k)-factor(normal(subs(n=n+1,f)/f,expanded)):" ",&*& -%\"hG6$,&%\"nG\"\"\"\"\"\"F)%\"kGF)-F%6$F(F+!\"\"F)-%'factorG6#-%'nor malG6$*&-%%subsG6$/F(,&F(F)\"\"\"F)%\"fGF)F " 0 "" {XPPEDIT 19 1 "h(n,k+1)/h(n,k)-factor(normal( subs(k=k+1,f)/f,expanded)):" ",&*&-%\"hG6$%\"nG,&%\"kG\"\"\"\"\"\"F*F* -F%6$F'F)!\"\"F*-%'factorG6#-%'normalG6$*&-%%subsG6$/F),&F)F*\"\"\"F*% \"fGF*F " 0 "" {MPLTEXT 1 0 34 " sys:=collect(map(numer,\{\"\",\"\}),h);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG<$,&*&,**$%\"nG\"\"$!\"\"*&F*\"\"#%\"kG\"\"\"F+*&F*F0F/F. !\"$*$F/F+F0F0-%\"hG6$F*F/F0F0*&,*F3F0*$F/F.F+F/F+F0F0F0-F56$F*,&F/F0F 0F0F0F0,&*&,6F)F,*$F*F.F2F-F+F*F2*&F*F0F/F0\"\"'F1F2F,F0F/F+F9F2F3F0F0 -F56$,&F*F0F0F0F/F0F0*&,*F)F0F@F+F*F+F0F0F0F4F0F0" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 52 "Summing, we get a recurrence for the Franel number s " }{XPPEDIT 18 0 "f[n]" "&%\"fG6#%\"nG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sys:=sum_of_sys(sys,k=-infinity..in finity,natural_boundaries);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG <#,**&,*!#c\"\"\"%\"nG!$O\"*$F+\"\"#!$/\"*$F+\"\"$!#CF*-%\"hG6#F+F*F** &,*F0F1F-\"#AF+\"#^\"#OF*F*-F46#,&F+F*F1F*F*F**&,*F0!#X!$S#F*F-FAF+!$> %F*-F46#,&F+F*F*F*F*F**&,*F0!#=!$[\"F*F-!$9\"F+!$K#F*-F46#,&F+F*F.F*F* F*" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "System for the Right-Hand Product" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Let us multiply the " }{XPPEDIT 18 0 "f[k]" "&%\"fG6#%\"kG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "binomial(n,k)*binomial(n+k,k)" "*&-%)binomialG6$%\"nG%\"kG\"\"\"-F$ 6$,&F&F(F'F(F'F(" }{TEXT -1 81 ". To do so, we prepare two systems of recurrence equations, one for each factor." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 31 "The following system describes " }{XPPEDIT 18 0 "f[k]" "&%\"fG6#%\"kG" }{TEXT -1 28 ", which is independent from " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "sys1:=\{h(n+1,k)-h(n,k)\} union eval(subs(h=proc(k) h (n,k) end,subs(n=k,sys)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%sys1G <$,&-%\"hG6$,&%\"nG\"\"\"F,F,%\"kGF,-F(6$F+F-!\"\",**&,*!#cF,F-!$O\"*$ F-\"\"#!$/\"*$F-\"\"$!#CF,F.F,F,*&,*F9F:F6\"#AF-\"#^\"#OF,F,-F(6$F+,&F -F,F:F,F,F,*&,*F9!#X!$S#F,F6FGF-!$>%F,-F(6$F+,&F-F,F,F,F,F,*&,*F9!#=!$ [\"F,F6!$9\"F-!$K#F,-F(6$F+,&F-F,F7F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "We next obtain a system that describes " }{XPPEDIT 18 0 " binomial(n,k)*binomial(n+k,k)" "*&-%)binomialG6$%\"nG%\"kG\"\"\"-F$6$, &F&F(F'F(F'F(" }{TEXT -1 30 ", the weight to multiply with:" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=binomial(n,k)*binomial(n+k, k):" ">%\"fG*&-%)binomialG6$%\"nG%\"kG\"\"\"-F&6$,&F(F*F)F*F)F*" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "h(n+1,k)/h(n,k)-factor(normal( subs(n=n+1,f)/f,expanded)):" ",&*&-%\"hG6$,&%\"nG\"\"\"\"\"\"F)%\"kGF) -F%6$F(F+!\"\"F)-%'factorG6#-%'normalG6$*&-%%subsG6$/F(,&F(F)\"\"\"F)% \"fGF)F " 0 "" {XPPEDIT 19 1 "h( n,k+1)/h(n,k)-factor(normal(subs(k=k+1,f)/f,expanded)):" ",&*&-%\"hG6$ %\"nG,&%\"kG\"\"\"\"\"\"F*F*-F%6$F'F)!\"\"F*-%'factorG6#-%'normalG6$*& -%%subsG6$/F),&F)F*\"\"\"F*%\"fGF*F " 0 "" {MPLTEXT 1 0 35 "sys2:=collect(map(numer,\{\"\",\"\}),h); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys2G<$,&*&,**$%\"nG\"\"#!\"\"F *F,%\"kG\"\"\"*$F-F+F.F.-%\"hG6$F*F-F.F.*&,(F/F.F-F+F.F.F.-F16$F*,&F-F .F.F.F.F.,&*&,(F*F,F,F.F-F.F.-F16$,&F*F.F.F.F-F.F.*&,(F*F.F.F.F-F.F.F0 F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We finally perform the pr oduct by a call to " }{HYPERLNK 17 "Mgfun[`sys*sys`]" 2 "Mgfun[`sys*sy s`]" "" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "s ys:=`sys*sys`(sys1,sys2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG<$ ,&*&,(%\"nG!\"\"F*\"\"\"%\"kGF*F+-%\"hG6$F)F,F+F+*&,(F)F+F+F+F,F*F+-F. 6$,&F)F+F+F+F,F+F+,**&,jnF,!$s'F)\"$s'*$F,\"\"'!$s#*$F)\"\"#\"$C#*&F)F +F,F+\"%/B*$F,F>!%GD*&F)F>F,F+\"%3;*&F)F>F,F>\"%kE*&F)F+F,F>\"%[I*$F, \"\"$!%;Q*$F,\"\"%!%gH*&F,FNF)F>\"$+'*&F,\"\"&F)F+\"#s*$F)FN!$!G*$F)FK !$S)*&F,FKF)F+\"%W>*&F)FSF,F+FT*$F)F;\"#c*&F)FNF,F>!$%Q*&F)FKF,F+!%o8* $F)FS\"$o\"*&F)F;F,F+\"#C*&F)FKF,F>!$o(*&F)FNF,F+!$C'*&F,FKF)FN!#s*&F, FKF)FK!$W\"*&F,FNF)F+FQ*$F,\"\"(!#C*&F,FKF)F>\"%s=*$F,FS!%[7*&F,FSF)F> FTF+F-F+F+*&,Z\"%!)GF+F,\"&[<\"F)!%?>F:FQF=!%!o\"F@!%s_FB\"&K,#FD!%`[F F!%7bFH!%_dFJ\"&<)=FM\"&s.\"FPFXFR!#!*FU\"$S#FW\"$![FY!%=JFhnF_qFjn\"$ Q)F`oF`qFbo\"$>%Fdo\"#XFfo\"#!*FhoFXFioFdqF\\p!%tIF^p\"%uLF`pF^qF+-F.6 $F),&F,F+F+F+F+F+*&,JFB!&%QF+F+F+*&,2FB!%%=&F:!#_Fio!\"$FJ!%vOFM!%S:F ,!%'*R!%'H\"F+F^p!$#QF+-F.6$F),&F,F+FKF+F+F+" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Recurrence for the Right-Hand Side" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "We obtain a recurrence for the right-hand side \+ by applying " }{HYPERLNK 17 "Mgfun[sum_of_sys]" 2 "Mgfun[sum_of_sys]" "" }{TEXT -1 48 " on the system computed in the previous section." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sum_of_sys(sys,k=-infinity.. infinity,takayama_algo);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<#,,*&,0!$ 7\"\"\"\"%\"nG!$m%*$F)\"\"#!$v(*$F)\"\"%!$&H*$F)\"\"$!$b'*$F)\"\"'!\"' *$F)\"\"&!#nF(-%\"hG6#F)F(F(*&,0F1\"&Uq$\"&!G9F(F4\"$/#F.\"&6O\"F7\"%7 EF+\"&&obF)\"&mR%F(-F;6#,&F)F(F(F(F(F(*&,0F4F5F7\"$8\"F.\"$q)F1\"%&\\$ F+\"%+xF)\"%%y)\"%KSF(F(-F;6#,&F)F(F/F(F(F(*&,0!&+#))F(F1!&y@*F+!'0#*= F4!$/#F7!%3NF)!'MF?F.!&6[#F(-F;6#,&F)F(F2F(F(F(*&,.F1!&wz&!&Sy(F(F+!'X /9F)!''3n\"F.!&v<\"F7!$U*F(-F;6#,&F)F(F,F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "This recurrence is different from the one obtained f or the left-hand side." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "r ec[right]:=op(collect(\",h,factor));" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#>&%$recG6#%&rightG,,*,,&%\"nG\"\"#\"\"(\"\"\"F.,&F+F,\"\"$F.F.,&F+F. F0F.F.,**$F+F0\"#^*$F+F,\"$p%F+\"%=9\"%+9F.F.-%\"hG6#F1F.!\"\"*,F/F.,& F+F0F-F.F.F1F.,&F+F.\"\"%F.F0-F:6#F?F.F.*,,&F+F0\"\")F.F.F*F.,&F+F.F,F .F.,&F+F.F.F.F0-F:6#F+F.F<*(,&F+F,\"\"&F.F.,,*$F+F@\"$r%F3\"%5ZF5\"&8s \"F+\"&!>F\"&ob\"F.F.-F:6#FFF.F<*,F*F.F/F.FFF.,*F3F4F5\"$'HF+\"$`&\"$S $F.F.-F:6#FGF.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "We thus need \+ more work to prove that both sides agree." }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 62 "Final Proof of the Identity and of the Second Order Rec urrence" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "h[ n]" "&%\"hG6#%\"nG" }{TEXT -1 95 " be any solution of the second order recurrence which has been obtained for the left-hand side:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rec[left];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&%\"nG\"\"\"F'F'\"\"$-%\"hG6#F&F'F'*&,&F&F'\"\" #F'F(-F*6#F-F'F'*(,&F&F.F(F'F',(*$F&F.\"# " 0 "" {MPLTEXT 1 0 41 "h(n+2)=collect(solve(\",h(n+2)),h,normal);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#,&%\"nG\"\"\"\"\"#F),&*(,**$F (\"\"$F)*$F(F*F/F(F/F)F)F),*F.F)F0\"\"'F(\"#7\"\")F)!\"\"-F%6#F(F)F5*( ,*F.\"#MF0\"$`\"F(\"$J#\"$<\"F)F)F1F5-F%6#,&F(F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(n=n+1,\");" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"hG6#,&%\"nG\"\"\"\"\"$F),&*(,**$,&F(F)F)F)F*F)*$ F/\"\"#F*F(F*\"\"%F)F),*F.F)F0\"\"'F(\"#7\"#?F)!\"\"-F%6#F/F)F7*(,*F. \"#MF0\"$`\"F(\"$J#\"$[$F)F)F3F7-F%6#,&F(F)F1F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(n=n+1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#,&%\"n G\"\"\"\"\"%F),&*(,**$,&F(F)\"\"#F)\"\"$F)*$F/F0F1F(F1\"\"(F)F),*F.F)F 2\"\"'F(\"#7\"#KF)!\"\"-F%6#F/F)F8*(,*F.\"#MF2\"$`\"F(\"$J#\"$z&F)F)F4 F8-F%6#,&F(F)F1F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Then, " } {XPPEDIT 18 0 "h[n]" "&%\"hG6#%\"nG" }{TEXT -1 52 " also solves the re currence for the right-hand side:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "collect(subs(\",\"\",\"\"\",rec[right]),h,normal);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 241 "At this point, we have proved that both sides of the equ ation satisfy the same recurrence of order 4. To prove the announced \+ equality, we simply need to check 4 initial conditions, since the lead ing coefficient of the recurrence of order 4," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "coeff(rec[right],h(n+3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,&%\"nG\"\"#\"\"(\"\"\"F),&F&F'\"\"$F)F),&F&F)F+F)F ),**$F&F+\"#^*$F&F'\"$p%F&\"%=9\"%+9F)F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "never vanishes for non-negative " }{XPPEDIT 18 0 "n" " I\"nG6\"" }{TEXT -1 32 ". Now the proof of the identity" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "eq:=Sum(binomial(n,k)^2*binomial(n+k, k)^2,k=0..n)=Sum(binomial(n,k)*binomial(n+k,k)*Sum(binomial(k,j)^3,j=0 ..k),k=0..n):" ">%#eqG/-%$SumG6$*&-%)binomialG6$%\"nG%\"kG\"\"#-F*6$,& F,\"\"\"F-F2F-\"\"#/F-;\"\"!F,-F&6$*(-F*6$F,F-F2-F*6$,&F,F2F-F2F-F2-F& 6$*$-F*6$F-%\"jG\"\"$/FD;F6F-F2/F-;F6F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "simplify follows from" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eval(subs(n=0,Sum=add,eq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\"F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eval(subs(n=1,Sum=add,eq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"& F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eval(subs(n=2,Sum=add ,eq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"#tF$" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 27 "eval(subs(n=3,Sum=add,eq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"%X9F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "T herefore, the Ap\351ry numbers also satisfy the announced second order recurrence." }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Computation of " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" }{TEXT -1 48 " Using Sta ndard Maple and the Holonomic Approach" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Standard Maple" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Ma ple has numerical routines for almost all special functions it knows a bout. Here is the corresponding calculation for " }{XPPEDIT 18 0 "Zet a(3)" "-%%ZetaG6#\"\"$" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ti[0]:=time():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Z3[standard]:=evalf(Zeta(3),332);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%#Z3G6#%)standardG$\"g_l`yBvt*\\R'ydr9q06=mT\\)pLANg( ymD`F)R?5\"y4rV:!))fD:d50%=*z7l9b'4y&4f6H.Tm4RchRC%\\UA_v:>>n1g$\\HyTR Xh,!z?PoRwP@$Gt')*f*>\\=%>2E&zd\"*>Y\"e_L4O(eX'=!48jy0#Q=Mb:F#z\")))\\ SBH')\\w!**\\9^h\"Q(*R&G%ffJ!p0-7!$J$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ti[standard]:=time()-ti[0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#tiG6#%)standardG$\"&:$Q!\"$" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 18 "Holonomic Approach" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We compute an approximation of " }{XPPEDIT 18 0 "Zeta(3) " "-%%ZetaG6#\"\"$" }{TEXT -1 61 " using Ap\351ry's recurrence. More \+ precisely, we compute it as " }{XPPEDIT 18 0 "a[200]/b[200]" "*&&%\"aG 6#\"$+#\"\"\"&%\"bG6#\"$+#!\"\"" }{TEXT -1 18 ". (Remember that " } {XPPEDIT 18 0 "a[n]/b[n]" "*&&%\"aG6#%\"nG\"\"\"&%\"bG6#F&!\"\"" } {TEXT -1 10 " tends to " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" } {TEXT -1 2 ".)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ti[0]:=ti me():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N:=200:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "To do so, we use the " }{HYPERLNK 17 "gfu n" 2 "gfun" "" }{TEXT -1 174 " package by Salvy and Zimmermann (Salvy, Bruno and Zimmermann, Paul (1994): Gfun: a Maple package for the mani pulation of generating and holonomic functions in one variable, " } {TEXT 261 25 "ACM Trans. Math. Software" }{TEXT -1 2 ", " }{TEXT 274 2 "20" }{TEXT -1 13 "(2):163-177)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7V%(La placeG%.algebraicsubsG%.algeqtodiffeqG%.algeqtoseriesG%.algfuntoalgeqG %&borelG%.cauchyproductG%.diffeq*diffeqG%.diffeq+diffeqG%2diffeqtohomd iffeqG%,diffeqtorecG%)guesseqnG%(guessgfG%0hadamardproductG%0holexprto diffeqG%)invborelG%,listtoalgeqG%-listtodiffeqG%0listtohypergeomG%+lis ttolistG%.listtoratpolyG%*listtorecG%-listtoseriesG%5listtoseries/Lapl aceG%1listtoseries/egfG%4listtoseries/lgdegfG%4listtoseries/lgdogfG%1l isttoseries/ogfG%4listtoseries/revegfG%4listtoseries/revogfG%,maxdegco effG%*maxdegeqnG%,maxordereqnG%,mindegcoeffG%*mindegeqnG%,minordereqnG %*optionsgfG%,poltodiffeqG%)poltorecG%/ratpolytocoeffG%(rec*recG%(rec+ recG%,rectodiffeqG%,rectohomrecG%*rectoprocG%.seriestoalgeqG%/seriesto diffeqG%2seriestohypergeomG%-seriestolistG%0seriestoratpolyG%,seriesto recG%/seriestoseriesG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 262 4 "gfun" }{TEXT -1 79 " package provides us with a routine f or transforming a recurrence equation like" }}}{EXCHG {PARA 0 "> " 0 " " {XPPEDIT 19 1 "eq:=n^3*u(n)-(34*n^3-51*n^2+27*n-5)*u(n-1)+(n-1)^3*u( n-2):" ">%#eqG,(*&%\"nG\"\"$-%\"uG6#F&\"\"\"F+*&,**&\"#MF+*$F&\"\"$F+F +*&\"#^F+*$F&\"\"#F+!\"\"*&\"#FF+F&F+F+\"\"&F6F+-F)6#,&F&F+\"\"\"F6F+F 6*&,&F&F+\"\"\"F6\"\"$-F)6#,&F&F+\"\"#F6F+F+" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 90 "into a procedure. Each of the following procedures enc odes the calculation of a sequence " }{XPPEDIT 18 0 "u[n]" "&%\"uG6#% \"nG" }{TEXT -1 23 " given by the equation " }{XPPEDIT 18 0 "eq" "I#eq G6\"" }{TEXT -1 24 " and its initial values " }{XPPEDIT 18 0 "u[0]" "& %\"uG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[1]" "&%\"uG6#\"\" \"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:=re ctoproc(\{eq,u(0)=0,u(1)=6\},u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%\"AG:6#%\"nG6&%\"iG%#u0G%#u1G%#u2G6\"E\\s#\"\"\"\"\"'\"\"!F1C&>8%F1 >8&F0?(8$\"\"#F/,&9$F/!\"\"F/%%trueGC%>8',$*&,(F4FF4F6 >F6F@,$*&,(F4F " 0 "" {MPLTEXT 1 0 38 "B:=rectopr oc(\{eq,u(0)=1,u(1)=5\},u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\" BG:6#%\"nG6&%\"iG%#u0G%#u1G%#u2G6\"E\\s#\"\"\"\"\"&\"\"!F/C&>8%F/>8&F0 ?(8$\"\"#F/,&9$F/!\"\"F/%%trueGC%>8',$*&,(F4FF4F6>F6F@,$* &,(F4F " 0 "" {MPLTEXT 1 0 8 "a:=A(N);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"]^m*HbNsb*zn 4KF)QKPAj@Xex]&G%y`$fh&Q!y?!ez5!pY0N)\\i0hqbzyeMh(>\\o%o(fXN9LQ5C$*4!H CkZ0sh\\$))p_kRtK+'*G;OG&34:%)3K6!z+v?ca&Q#>@vw>X'*R\\_'=b>NuiNZSKa>Z0 BS1HDNfy]T(Gs0yRJ2'4d<]syf&Qqpb*p2fm@m8sC\")fb-ie4=t(*3'*=MvE-A&\\c'H4 %)>Dg)[pY8&oR5w*>^%pJeX:SefU;r\\QB'4T/Oy,:g#G#ps4)[z3'RGT*QShXAWRyPo%* oq#pt'e!HstpN(Q!\\0\"=e['z/a_9-7F\"_[l++++#RfsaYgl=SYT\\5%>-![;W$RWZ0f a&pBU[2/!QK1XVx$4m'fyY@'\\/v!=F.qRy8][]x%*yYoN%*\\,;\\f>z@6%oo*3sr9$oh *fz5P&=>$))=4#>\"\\*))f*G*QLZuMT*=9Xh)Q[Es\"=Mo%)*)[<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b:=B(N);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"bG\"j]lDu?l%eHmOf:Ow9LZb%f'p23Li]gMN1\\]$o>0QBk&*)3 !fPMA.M@fdRZ![0y8N9<#)fOuuBf/2-9X;Yt=;]\"Hb(Gw#*4XP7\"QM9-hXFIII$p[)\\ '*3zCyC*3>z![\"QA#>1aUt6?Am%=+VI(G]azsa'Qf%e?o,Ek#Rm\\4/!H\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 263 8 "A priori" }{TEXT -1 51 ", it is no t clear how many digits we can guarantee." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "P:=round(evalf(ln(a)/ln(10))*1.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG\"$K$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Z3[holonomy]:=evalf(a/b,P);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>&%#Z3G6#%)holonomyG$\"g_l`yBvt*\\R'ydr9q06=mT\\)pLANg(ymD`F)R?5\"y4r V:!))fD:d50%=*z7l9b'4y&4f6H.Tm4RchRC%\\UA_v:>>n1g$\\HyTRXh,!z?PoRwP@$G t')*f*>\\=%>2E&zd\"*>Y\"e_L4O(eX'=!48jy0#Q=Mb:F#z\")))\\SBH')\\w!**\\9 ^h\"Q(*R&G%ffJ!p0-7!$J$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The ti me used is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ti[holonomy]: =time()-ti[0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#tiG6#%)holonomyG $\"%'G(!\"$" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Comparison" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The holonomic approach is several \+ time faster." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ti[standard ]/ti[holonomy];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+W`re_!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Moreover, this ratio would increa se with the accuracy of the calculations. In this session, we have ob tained " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" }{TEXT -1 18 " up \+ to 332 digits." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Z3[standa rd]-Z3[holonomy];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "However, " }{TEXT 268 5 "Maple" }{TEXT -1 26 " would be able to compute " }{XPPEDIT 18 0 "Zeta(z)" "-%%ZetaG6#% \"zG" }{TEXT -1 5 " for " }{TEXT 267 3 "any" }{TEXT -1 15 " complex va lue " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 46 ", while the holonomi c approach only works for " }{XPPEDIT 18 0 "z=3" "/%\"zG\"\"$" }{TEXT -1 1 "." }}}}}}{MARK "0 1 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 }