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{SECT 0 {EXCHG {PARA 202 "" 1 "" {TEXT 206 66 "An Algolib-aided Version of Apery's Proof of the Irrationality of " }{XPPEDIT 18 0 "zeta(3);" "6#-%%zetaG6#\"\"$" }}
}
{EXCHG {PARA 203 "" 0 "" {TEXT 207 11 "Bruno Salvy" }}
{PARA 203 "" 0 "" {TEXT 207 15 "(March 4, 2003)" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 26 "Apery proved in 1978 that " }{XPPEDIT 18 0 "zeta(3) = Sum(1/(k^3),k = 1 .. infinity);" "6#/-%%zetaG6#\"\"$-%$SumG6$*&\"\"\"F,*$%\"kGF'!\"\"/F.;F,%)infinityG" }{TEXT 208 198 " is irrational. We give a short version of Apery's proof that uses several tools from Algolib: gfun, Mgfun and equivalent. We only prove irrationality here and do not compute irrationality measures." }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 56 "The starting point is the definition of three sequences:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 42 "c[n,k]:=binomial(n,k)^2*binomial(n+k,k)^2;" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#>&%\"cG6$%\"nG%\"kG*&)-%)binomialGF&\"\"#\"\"\")-F,6$,&F'F.F(F.F(F-F." }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 25 "a[n]:=Sum(c[n,k],k=0..n);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#%\"nG-%$SumG6$*&)-%)binomialG6$F'%\"kG\"\"#\"\"\")-F.6$,&F'F2F0F2F0F1F2/F0;\"\"!F'" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 109 "b[n]:='a[n]'*Sum(1/k^3,k=1..n)+Sum(Sum((-1)^(m+1)*c[n,k]/2/m^3/binomial(n,m)/binomial(n+m,m),m=1..k),k=1..n);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#%\"nG,&*&&%\"aGF&\"\"\"-%$SumG6$*&F,F,*$)%\"kG\"\"$F,!\"\"/F3;F,F'F,F,-F.6$-F.6$,$*&#F,\"\"#F,*.)F5,&%\"mGF,F,F,F,-%)binomialG6$F'F3F?-FE6$,&F'F,F3F,F3F?FC!\"$-FE6$F'FCF5-FE6$,&F'F,FCF,FCF5F,F,/FC;F,F3F6F," }}
}
{PARA 204 "" 0 "" }
{SECT 0 {PARA 206 "" 1 "" {TEXT 210 3 "1. " }{XPPEDIT 18 0 "limit(b[n]/a[n],n = infinity) = zeta(3);" "6#/-%&limitG6$*&&%\"bG6#%\"nG\"\"\"&%\"aGF*!\"\"/F+%)infinityG-%%zetaG6#\"\"$" }}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 59 "The first part of the right-hand side of the definition of " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 32 " provides the desired limit for " }{XPPEDIT 18 0 "b[n]" "6#&%\"bG6#%\"nG" }{TEXT 208 1 "/" }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT 208 81 ". The proof consists in showing that the second part tends to 0, when divided by " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 208 5 ". Let" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 37 "U:=m^3*binomial(n,m)*binomial(n+m,m);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#>%\"UG*()%\"mG\"\"$\"\"\"-%)binomialG6$%\"nGF'F)-F+6$,&F-F)F'F)F'F)" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 34 "What we are going to show is that " }{XPPEDIT 18 0 "n^2 <= U;" "6#1*$%\"nG\"\"#%\"UG" }{TEXT 208 25 ". Thus the second sum in " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 14 " is bounded by" }{XPPEDIT 18 0 "Sum(Sum(c[n,k]/2/(n^2),m = 1 .. k),k = 1 .. n) <= Sum(k*c[n,k]/2/(n^2),k = 1 .. n);" "6#1-%$SumG6$-F%6$*(&%\"cG6$%\"nG%\"kG\"\"\"\"\"#!\"\"*$F-F0F1/%\"mG;F/F./F.;F/F--F%6$**F.F/F*F/F0F1F2F1F6" }{TEXT 208 20 ", itself bounded by " }{XPPEDIT 18 0 "a[n]/(2*n);" "6#*&&%\"aG6#%\"nG\"\"\"*&\"\"#F(F'F(!\"\"" }{TEXT 208 28 ", whence the desired limit. " }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 79 "The following simple way of proving this was suggested to us by Philippe Dumas:" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 40 "For m=n, we have the desired inequality:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 12 "eval(U,m=n);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#*&)%\"nG\"\"$\"\"\"-%)binomialG6$,$*&\"\"#F'F%F'F'F%F'" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 24 "Otherwise, we have that " }{XPPEDIT 18 0 "binomial(n,m) <= binomial(n+m,m);" "6#1-%)binomialG6$%\"nG%\"mG-F%6$,&F'\"\"\"F(F,F(" }{TEXT 208 39 ", and the conclusion follows since for " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT 208 15 " between 1 and " }{XPPEDIT 18 0 "n-1;" "6#,&%\"nG\"\"\"F%!\"\"" }{TEXT 208 10 ", one has " }{XPPEDIT 18 0 "n <= binomial(n,m);" "6#1%\"nG-%)binomialG6$F$%\"mG" }{TEXT 208 1 "." }}
}
}
{SECT 0 {PARA 206 "" 1 "" {TEXT 210 3 "2. " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 210 27 " is a positive integer and " }{XPPEDIT 18 0 "d[n]^3*b[n];" "6#*&&%\"dG6#%\"nG\"\"$&%\"bGF&\"\"\"" }{TEXT 210 22 " is an integer, where " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#%\"nG" }{TEXT 210 21 " is the lcm of 1,...," }{XPPEDIT 18 0 "n;" "6#%\"nG" }}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 56 "The first assertion comes from binomials being integers." }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 83 "The second one is obtained by showing that each summand in the double sum defining " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 28 " has a denominator dividing " }{XPPEDIT 18 0 "d[k];" "6#&%\"dG6#%\"kG" }{TEXT 208 39 ". This is done in two steps as follows:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 156 "(binomial(n,k)^2*binomial(n+k,k)^2/m^3/binomial(n,m)/binomial(n+m,m))=binomial(n,k)*binomial(n+k,k)*binomial(n-m,n-k)*binomial(n+k,k-m)/m^3/binomial(k,m)^2;" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#/*,-%)binomialG6$%\"nG%\"kG\"\"#-F&6$,&F(\"\"\"F)F.F)F*%\"mG!\"$-F&6$F(F/!\"\"-F&6$,&F(F.F/F.F/F3*.F%F.F+F.-F&6$,&F(F.F/F3,&F(F.F)F3F.-F&6$F-,&F)F.F/F3F.F/F0-F&6$F)F/!\"#" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 6 "Proof:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 29 "convert(lhs(%)/rhs(%),GAMMA);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 42 "and then it is sufficient to observe that " }{XPPEDIT 18 0 "m*binomial(k,m);" "6#*&%\"mG\"\"\"-%)binomialG6$%\"kGF$F%" }{TEXT 208 9 " divides " }{XPPEDIT 18 0 "d[k];" "6#&%\"dG6#%\"kG" }{TEXT 208 2 ". " }}
}
}
{SECT 0 {PARA 206 "" 1 "" {TEXT 210 8 "3. Both " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 210 5 " and " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 210 35 " satisfy a simple linear recurrence" }}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 18 "This is where the " }{HYPERLNK 211 "Mgfun" 2 "Mgfun" "" }{TEXT 208 24 " package will be useful." }}
}
{SECT 0 {PARA 208 "" 1 "" {TEXT 212 24 "Recurrence satisfied by " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 58 "CT:=Mgfun[creative_telescoping](c[n,k],n::shift,k::shift);" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#>%#CTG7$,(*&,**&\"\"$\"\"\")%\"nG\"\"#F+!\"\"*&F*F+F-F+F/*$)F-F*F+F/F+F/F+-%#_fG6$F-%\"kGF+F+*&,*F1F/*&\"\"'F+F,F+F/*&\"#7F+F-F+F/\"\")F/F+-F46$,&F-F+F.F+F6F+F+*&,**&\"#MF+F2F+F+*&\"$`\"F+F,F+F+*&\"$J#F+F-F+F+\"$<\"F+F+-F46$,&F-F+F+F+F6F+F+,$*.\"\"%F+F6FO,,F;F+*&FOF+F,F+F+F=F+*&F*F+F6F+F+*&F.F+)F6F.F+F/F+,&*&F.F+F-F+F+F*F+F+,@FOF+*(\"#=F+F,F+F6F+F/*(FOF+F2F+F6F+F/*(FOF+F-F+)F6F*F+F/*(F:F+F,F+FTF+F+F;F+*&F<F+F6F+F/*&F:F+FfnF+F/*$)F-FOF+F+*&F:F+F2F+F+*&\"#8F+F,F+F+*&F^oF+FTF+F+*$)F6FOF+F+*(\"#EF+F-F+F6F+F/*(FYF+F-F+FTF+F+F/F3F+F/" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 100 "This result means that applying the recurrence operator given by the first element of this list to  " }{XPPEDIT 18 0 "c[n,k];" "6#&%\"cG6$%\"nG%\"kG" }{TEXT 208 26 ", one gets the difference " }{XPPEDIT 18 0 "g(n,k)-g(n,k+1);" "6#,&-%\"gG6$%\"nG%\"kG\"\"\"-F%6$F',&F(F)F)F)!\"\"" }{TEXT 208 8 ", where " }{XPPEDIT 18 0 "g;" "6#%\"gG" }{TEXT 208 49 " is the second element of this list. Explicitly, " }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 54 "eval(CT[1],_f=proc(N,K) 'c[N,K]' end)=g(n,k)-g(n,k+1);" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#/,(*&,**&\"\"$\"\"\")%\"nG\"\"#F)!\"\"*&F(F)F+F)F-*$)F+F(F)F-F)F-F)&%\"cG6$F+%\"kGF)F)*&,*F/F-*&\"\"'F)F*F)F-*&\"#7F)F+F)F-\"\")F-F)&F26$,&F+F)F,F)F4F)F)*&,**&\"#MF)F0F)F)*&\"$`\"F)F*F)F)*&\"$J#F)F+F)F)\"$<\"F)F)&F26$,&F+F)F)F)F4F)F),&-%\"gGF3F)-FM6$F+,&F4F)F)F)F-" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 4 "with" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 36 "g(n,k)=subs(_f(n,k)='c[n,k]',CT[2]);" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#/-%\"gG6$%\"nG%\"kG,$*.\"\"%\"\"\"F(F+,,*&\"#7F,F'F,F,*&F+F,)F'\"\"#F,F,\"\")F,*&\"\"$F,F(F,F,*&F2F,)F(F2F,!\"\"F,,&*&F2F,F'F,F,F5F,F,,@F+F,*(\"#=F,F1F,F(F,F8*(F+F,)F'F5F,F(F,F8*(F+F,F'F,)F(F5F,F8*(\"\"'F,F1F,F7F,F,F.F,*&F/F,F(F,F8*&FCF,FAF,F8*$)F'F+F,F,*&FCF,F?F,F,*&\"#8F,F1F,F,*&FJF,F7F,F,*$)F(F+F,F,*(\"#EF,F'F,F(F,F8*(F=F,F'F,F7F,F,F8&%\"cGF&F,F8" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 87 "We check this formula by computing the ratio left-hand side divided by right-hand side:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 36 "ct:=eval(CT,_f=unapply(c[n,k],n,k)):" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 48 "normal(expand(ct[1]/(ct[2]-subs(k=k+1,ct[2]))));" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 31 "Now, summing this identity for " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT 208 11 " from 0 to " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 208 60 "+2, we get that the right-hand side telescopes, leaving out " }{XPPEDIT 18 0 "g(n,0)-g(n,n+3);" "6#,&-%\"gG6$%\"nG\"\"!\"\"\"-F%6$F',&F'F)\"\"$F)!\"\"" }{TEXT 208 20 ". This in turn is 0:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 16 "eval(ct[2],k=0);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 27 "normal(limit(ct[2],k=n+3));" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 13 "The sequence " }{XPPEDIT 18 0 "c[n,k];" "6#&%\"cG6$%\"nG%\"kG" }{TEXT 208 10 " is 0 for " }{XPPEDIT 18 0 "k = n+1;" "6#/%\"kG,&%\"nG\"\"\"F'F'" }{TEXT 208 5 " and " }{XPPEDIT 18 0 "k = n+2;" "6#/%\"kG,&%\"nG\"\"\"\"\"#F'" }{TEXT 208 25 ", and thus its sum up to " }{XPPEDIT 18 0 "k=n+2" "6#/%\"kG,&%\"nG\"\"\"\"\"#F'" }{TEXT 208 13 " is equal to " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT 208 1 "." }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 38 "Thus we have proved that the sequence " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 208 42 " satisfies the following recurrence (with " }{XPPEDIT 18 0 "A = a;" "6#/%\"AG%\"aG" }{TEXT 208 1 ")" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 54 "collect(eval(CT[1],_f=proc(n,k) A(n) end),A,factor)=0;" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#/,(*&),&%\"nG\"\"\"F)F)\"\"$F)-%\"AG6#F(F)!\"\"*&),&F(F)\"\"#F)F*F)-F,6#F1F)F.*(,&*&F2F)F(F)F)F*F)F),(*&\"#<F))F(F2F)F)*&\"#^F)F(F)F)\"#RF)F)-F,6#F'F)F)\"\"!" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 13 "rec:=op(1,%):" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 62 "We check this recurrence by computing the first few values of " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 208 1 ":" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 44 "lista:=[seq(value(subs(n=i,a[n])),i=1..20)];" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#>%&listaG76\"\"&\"#t\"%X9\"&,I$\"'0!>)\")D3Y@\"*ltI%e\",DC\"zO;\"-0!\\3po%\"/tISOul8\"0D\")y$3wOS\"2HZc5]Ot?\"\"3DP)fRsNrk$\"5t!)>Vr*>d66\"\"6Xa5F=_/X.T$\"8DGE$H'R)>AX`5\"9&[@BWhh^Q$fsK\";DM:PZ#\\X_#*p<-\"\"<X/\\zNS@WD;QX?$\">,g$G%G.%*pEqC%445" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 68 "and then substituting these in the left-hand side of the recurrence:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 52 "seq(eval(rec,[n=i,A=proc(k) lista[k] end]),i=1..18);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "64\"\"!F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#" }}
}
}
{SECT 0 {PARA 208 "" 1 "" {TEXT 212 24 "Recurrence satisfied by " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 82 "First, we observe that the recurrence above seems to be satisfied by the sequence " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 1 ":" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 160 "for i to 10 do listb[i]:=lista[i]*add(1/k^3,k=1..i)+add(binomial(i,k)^2*binomial(i+k,k)^2/2*add((-1)^(m-1)/m^3/binomial(i,m)/binomial(i+m,m),m=1..k),k=1..i) od:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 51 "seq(eval(rec,[n=i,A=proc(k) listb[k] end]),i=2..7);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6(\"\"!F#F#F#F#F#" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 18 "In order to get a " }{TEXT 213 5 "proof" }{TEXT 208 46 " that the recurrence is actually satisfied by " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 190 ", we use a similar process based on creative telescoping. The situation is slightly more complicated, since it is necessary to obtain a system of recurrences satisfied by the summand first. " }}
}
{SECT 0 {PARA 209 "" 1 "" {TEXT 214 22 "Indefinite sum inside " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 130 "We start with the inner summand, for the indefinite sum of which we need to find description making creative telescoping possible:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 55 "d[n,m]:=(-1)^(m+1)/2/m^3/binomial(n,m)/binomial(n+m,m);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#>&%\"dG6$%\"nG%\"mG,$*&#\"\"\"\"\"#F,**)!\"\",&F(F,F,F,F,F(!\"$-%)binomialGF&F0-F46$,&F'F,F(F,F(F0F,F," }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 59 "We first get operators by definition of the indefinite sum:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 56 "Mgfun[dfinite_expr_to_sys](d[n,m],f(n::shift,m::shift));" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#<$,&*&,(%\"nG!\"\"%\"mG\"\"\"F*F(F*-%\"fG6$F'F)F*F**&,(F'F*F)F*F*F*F*-F,6$,&F'F*F*F*F)F*F*,&*&)F)\"\"$F*F+F*F**&,0*&F)F*)F'\"\"#F*F**$F:F*F**&F)F*F'F*F*F'F**&F;F*)F)F;F*F(*$F5F*F(F)F(F*-F,6$F',&F)F*F*F*F*F*" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 78 "firstoperators:=subs(k=k+1,subs(m=k,eval(%,f=proc(n,m) g(n,m)-g(n,m-1) end)));" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#>%/firstoperatorsG<$,&*&),&%\"kG\"\"\"F+F+\"\"$F+,&-%\"gG6$%\"nGF)F+-F/6$F1F*!\"\"F+F+*&,2*&)F1\"\"#F+F)F+F+*$F8F+F+*&F1F+F)F+F+F1F+*&F9F+)F)F9F+F4*$F(F+F4F*F4F+F4F+,&-F/6$F1,&F9F+F*F+F+F.F4F+F+,&*&,&F1F4F*F+F+F-F+F+*&,(F1F+F9F+F*F+F+,&-F/6$,&F1F+F+F+F)F+-F/6$FKF*F4F+F+" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 90 "We check that these operators cancel our sequence when evaluated at a special point (here " }{XPPEDIT 18 0 "n = 10,k = 6;" "6$/%\"nG\"#5/%\"kG\"\"'" }{TEXT 208 2 "):" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 66 "eval(firstoperators,g=proc(N,K) Sum(subs(n=N,d[n,m]),m=1..K) end):" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 13 "eval(%,n=10):" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 19 "value(subs(k=6,%));" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#<#\"\"!" }}
}
{SECT 0 {PARA 210 "" 0 "" {TEXT 215 48 "These operators do not generate a D-finite ideal" }}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 651 "In order to ensure the success of creative telescoping, we need to compute \"sufficiently many\" operators annihilating our sequence. This \"sufficiently many\" is formally defined in terms of the dimension of the ideal our operators generate. A D-finite ideal is an ideal such that the quotient of the algebra of operators with the ideal is finitely dimensional as a vector space. Informally, this amounts to saying that all the shifts of a sequence annihilated by the operators can be rewritten as linear combination of a finite number of them. Technically, D-finiteness can be checked by computing the Hilbert dimension of the ideal, which has to be 0." }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 73 "sys:=subs([seq(seq(g(n+i,k+j)=Sn^i*Sk^j,j=0..2),i=0..1)],firstoperators);" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#>%$sysG<$,&*&,&%\"nG!\"\"%\"kG\"\"\"F,,&%#SkGF,F,F*F,F,*&,(F)F,\"\"#F,F+F,F,,&*&%#SnGF,F.F,F,F4F*F,F,,&*&),&F+F,F,F,\"\"$F,F-F,F,*&,2*&)F)F1F,F8F,F,*$F=F,F,*&F)F,F8F,F,F)F,*&F1F,)F8F1F,F**$F7F,F*F+F*F,F*F,,&*$)F.F1F,F,F.F*F,F," }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 101 "Groebner[hilbertdim](sys,Groebner[termorder](Ore_algebra[shift_algebra]([Sn,n],[Sk,k]),tdeg(Sn,Sk)));" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 52 "This would be zero in the case of a D-finite system." }}
}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 53 "More generators are obtained by creative telescoping:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 59 "CT2:=Mgfun[creative_telescoping](d[n,m],n::shift,m::shift);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#>%$CT2G7$,&-%#_fG6$%\"nG%\"mG!\"\"-F(6$,&F*\"\"\"F0F0F+F0,$*,\"\"#F0F+F0,(F*F0F+F,F0F0F0,(*$)F*F3F0F0*&F3F0F*F0F0F0F0F,F'F0F," }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 22 "Then, in order to sum " }{XPPEDIT 18 0 "d[n,m];" "6#&%\"dG6$%\"nG%\"mG" }{TEXT 208 70 " for m from 1 to k, we first find annihilators of the right-hand side:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 115 "Mgfun[dfinite_expr_to_sys](subs(_f(n,m)=d[n,m],m=k+1,CT2[2])-subs(_f(n,m)=d[n,m],m=1,CT2[2]),f(n::shift,k::shift));" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#<$,(*&,8\"\"&\"\"\"*(\"#@F()%\"nG\"\"#F(%\"kGF(!\"\"*(\"#6F()F,\"\"$F(F.F(F/*&\"#8F()F,\"\"%F(F(*&\"#KF(F2F(F(*&\"#AF(F,F(F(*&F'F(F.F(F/*(\"#<F(F,F(F.F(F/*&\"#QF(F+F(F(*&F-F()F,F'F(F(*(F-F(F6F(F.F(F/F(-%\"fG6$F,F.F(F(*&,8FAF(*&\"#FF(F6F(F(FCF(*(F*F(F2F(F.F(F(*&\"$W\"F(F2F(F(*&\"$y$F(F+F(F(*(\"#\")F(F+F(F.F(F(*(\"$N\"F(F,F(F.F(F(*&\"$'[F(F,F(F(\"$V#F(*&FQF(F.F(F(F(-FE6$,&F,F(F-F(F.F(F(*&,6\"$7\"F/*(F-F(F2F(F.F(F(*(\"#CF(F,F(F.F(F(*(\"#7F(F+F(F.F(F(*&F7F(FBF(F/*&\"$e\"F(F2F(F/*&\"$3$F(F+F(F/*&\"$'HF(F,F(F/*&\"#;F(F.F(F(*&\"#SF(F6F(F/F(-FE6$,&F,F(F(F(F.F(F(,(*&,2*&F3F()F.F-F(F(*$F2F(F(*(F-F(F+F(F.F(F/*&F3F(F+F(F(*(F3F(F,F(F.F(F/*(F-F(F,F(F_pF(F(*&F3F(F,F(F(F(F(F(FDF(F(*&,*F`pF/*&\"\"'F(F+F(F/*&F\\oF(F,F(F/\"\")F/F(FhoF(F(*&,*FapF(FcpF(FdpF/F^pF/F(-FE6$F,,&F.F(F(F(F(F(" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 90 "and then we get annihilators for the indefinite sum by substitution in the left-hand side:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 54 "ann:=eval(%,f=proc(N,K) unapply(CT2[1],n,m)(N,K) end);" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#>%$annG<$,(*&,8\"\"&\"\"\"*(\"#@F*)%\"nG\"\"#F*%\"kGF*!\"\"*(\"#6F*)F.\"\"$F*F0F*F1*&\"#8F*)F.\"\"%F*F**&\"#KF*F4F*F**&\"#AF*F.F*F**&F)F*F0F*F1*(\"#<F*F.F*F0F*F1*&\"#QF*F-F*F**&F/F*)F.F)F*F**(F/F*F8F*F0F*F1F*,&-%#_fG6$F.F0F1-FH6$,&F.F*F*F*F0F*F*F**&,8FCF**&\"#FF*F8F*F*FEF**(F,F*F4F*F0F*F**&\"$W\"F*F4F*F**&\"$y$F*F-F*F**(\"#\")F*F-F*F0F*F**(\"$N\"F*F.F*F0F*F**&\"$'[F*F.F*F*\"$V#F**&FWF*F0F*F*F*,&-FH6$,&F.F*F/F*F0F1-FH6$,&F.F*F5F*F0F*F*F**&,6\"$7\"F1*(F/F*F4F*F0F*F**(\"#CF*F.F*F0F*F**(\"#7F*F-F*F0F*F**&F9F*FDF*F1*&\"$e\"F*F4F*F1*&\"$3$F*F-F*F1*&\"$'HF*F.F*F1*&\"#;F*F0F*F**&\"#SF*F8F*F1F*,&FJF1FinF*F*F*,(*&,2*&F5F*)F0F/F*F**$F4F*F**(F/F*F-F*F0F*F1*&F5F*F-F*F**(F5F*F.F*F0F*F1*(F/F*F.F*FgpF*F**&F5F*F.F*F*F*F*F*FFF*F**&,*FhpF1*&\"\"'F*F-F*F1*&FfoF*F.F*F1\"\")F1F*FbpF*F**&,*FipF*F[qF*F\\qF1FfpF1F*,&-FH6$F.,&F0F*F*F*F1-FH6$FLFiqF*F*F*" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 52 "Again, we check that these operators are correct at " }{XPPEDIT 18 0 "n = 10,k = 6;" "6$/%\"nG\"#5/%\"kG\"\"'" }{TEXT 208 1 ":" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 56 "eval(ann,_f=proc(N,K) Sum(subs(n=N,d[n,m]),m=1..K) end):" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 13 "subs(n=10,%):" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 19 "value(subs(k=6,%));" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#<#\"\"!" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 50 "Finally, we put both sets of generators together: " }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 65 "ann2:=map(collect,ann union subs(g=_f,firstoperators),_f,factor);" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#>%%ann2G<&,,*&),&%\"nG\"\"\"\"\"#F+\"\"$F+-%#_fG6$F)%\"kGF+!\"\"*(,&*&F,F+F*F+F+F-F+F+,,*$)F*F,F+F+*&F*F+F1F+F2*&F-F+F*F+F+*$)F1F,F+F+F-F+F+-F/6$,&F*F+F+F+F1F+F+*&,2*&F-F+F<F+F2*$)F*F-F+F2*(F,F+F8F+F1F+F+*&F-F+F8F+F2*(F-F+F*F+F1F+F+*(F,F+F*F+F<F+F2F:F2F+F2F+-F/6$F*F1F+F+**F1F+F4F+,&F*F+F1F2F+-F/6$F?,&F1F+F+F+F+F+**F1F+F4F+FLF+-F/6$F*FOF+F2,(*&)FOF-F+FIF+F2*&,2*&F,F+)F1F-F+F+*&\"\")F+F<F+F+*&\"#6F+F1F+F+\"\"&F+*&F8F+F1F+F2*&F,F+F8F+F2F9F2F5F2F+FQF+F+**,&F,F+F1F+F+,(F*F+F,F+F1F+F+,(F*F+F1F2F+F2F+-F/6$F*F\\oF+F+,**(,&F5F+FhnF+F+,4*&F-F+)F*\"\"%F+F+*&\"#EF+FDF+F+*&FDF+F1F+F+*&\"#')F+F8F+F+*(\"\"(F+F8F+F1F+F+*&\"$G\"F+F*F+F+*(\"#<F+F*F+F1F+F+\"#rF+*&\"#8F+F1F+F+F+F.F+F2*(F4F+,4FeoF+*&\"#AF+FDF+F+FjoF2*(FhnF+F8F+F1F+F2*&\"#iF+F8F+F+*&\"#!)F+F*F+F+*(\"\"*F+F*F+F1F+F2\"#RF+*&F^pF+F1F+F2F+F=F+F+**FcoF+)F?F-F+,(F*F+F1F2F+F+F+FIF+F2**F4F+),&F*F+F-F+F-F+,(F*F+F-F+F1F+F+-F/6$FhqF1F+F+,**&,(F*F2F,F2F1F2F+F=F+F+*&FLF+FIF+F+*&,&F*F2F1F+F+FQF+F+*&F]oF+FMF+F+" }}
}
{SECT 0 {PARA 210 "" 0 "" {TEXT 215 21 "Now, this is D-finite" }}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 64 "sys:=subs([seq(seq(_f(n+i,k+j)=Sn^i*Sk^j,j=0..2),i=0..3)],ann2);" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#>%$sysG<&,(*$),&%\"kG\"\"\"F+F+\"\"$F+!\"\"*&,2*&\"\"#F+)F*F,F+F+*&\"\")F+)F*F1F+F+*&\"#6F+F*F+F+\"\"&F+*&)%\"nGF1F+F*F+F-*&F1F+F:F+F-*&F;F+F*F+F-*&F1F+F;F+F-F+%#SkGF+F+**,&F1F+F*F+F+,(F;F+F1F+F*F+F+,(F;F+F*F-F+F-F+)F?F1F+F+,:*&),&F;F+F1F+F,F+)%#SnGF1F+F-*(,&F>F+F,F+F+,,*$F:F+F+F=F-*&F,F+F;F+F+*$F5F+F+F,F+F+FJF+F+*&F,F+F5F+F-*$)F;F,F+F-*(F1F+F:F+F*F+F+*&F,F+F:F+F-*(F,F+F;F+F*F+F+*(F1F+F;F+F5F+F-FOF-F+F-*,F*F+FLF+,&F;F+F*F-F+FJF+F?F+F+**F*F+FLF+FYF+F?F+F-,,*&,(F;F-F1F-F*F-F+FJF+F+F;F+F*F-*&,&F;F-F*F+F+F?F+F+*(FBF+FJF+F?F+F+,**(,&F>F+F8F+F+,4*&F,F+)F;\"\"%F+F+*&\"#EF+FSF+F+*&FSF+F*F+F+*&\"#')F+F:F+F+*(\"\"(F+F:F+F*F+F+*&\"$G\"F+F;F+F+*(\"#<F+F;F+F*F+F+\"#rF+*&\"#8F+F*F+F+F+FIF+F-*(FLF+,4F_oF+*&\"#AF+FSF+F+FdoF-*(F8F+F:F+F*F+F-*&\"#iF+F:F+F+*&\"#!)F+F;F+F+*(\"\"*F+F;F+F*F+F-\"#RF+*&FhoF+F*F+F-F+FJF+F+*(F]oF+),&F;F+F+F+F,F+,(F;F+F*F-F+F+F+F-**FLF+),&F;F+F,F+F,F+,(F;F+F,F+F*F+F+)FJF,F+F+" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 101 "Groebner[hilbertdim](sys,Groebner[termorder](Ore_algebra[shift_algebra]([Sn,n],[Sk,k]),tdeg(Sn,Sk)));" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
}
}
}
{SECT 0 {PARA 209 "" 0 "" {TEXT 214 23 "Building up the summand" }}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 75 "From the (D-finite) system of operators annihilating the indefinite sum of " }{XPPEDIT 18 0 "d[n,m];" "6#&%\"dG6$%\"nG%\"mG" }{TEXT 208 82 ", we are now going to construct a system of operators annihilating the summand in " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 154 ", before applying creative telescoping. This last step will be possible because addition and product of D-finite systems remain D-finite. Thus we now add " }{XPPEDIT 18 0 "Sum(1/(m^3),m = 1 .. n);" "6#-%$SumG6$*&\"\"\"F'*$%\"mG\"\"$!\"\"/F);F'%\"nG" }{TEXT 208 17 " and multiply by " }{XPPEDIT 18 0 "c[n,k];" "6#&%\"cG6$%\"nG%\"kG" }{TEXT 208 71 ". This is achieved by first constructing a system annihilating the sum:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 37 "inhom:=(n+1)^3*(_f(n+1,k)-_f(n,k))-1;" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#>%&inhomG,&*&),&%\"nG\"\"\"F*F*\"\"$F*,&-%#_fG6$F)%\"kG!\"\"-F.6$F(F0F*F*F*F*F1" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 130 "(the system is composed of this equation, which we render homogeneous, and another one reflecting that the sum does not depend on " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT 208 53 "). Then we construct the system satisfied by the sum:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 67 "Mgfun[`sys+sys`](\{_f(n,k+1)-_f(n,k),subs(n=n+1,inhom)-inhom\},ann2);" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#<%,**&,(%\"nG!\"\"\"\"#F(%\"kGF(\"\"\"-%#_fG6$,&F'F+F+F+F*F+F+*&,&F'F+F*F(F+-F-6$F'F*F+F+*&,&F'F(F*F+F+-F-6$F',&F*F+F+F+F+F+*&,(F'F+F)F+F*F+F+-F-6$F/F8F+F+,(*&,**$)F*\"\"$F+F(*&FBF+)F*F)F+F(*&FBF+F*F+F(F+F(F+F2F+F+*&,2*&F)F+FAF+F+*&\"\")F+FDF+F+*&\"#6F+F*F+F+\"\"&F+*&)F'F)F+F*F+F(*&F)F+FOF+F(*&F'F+F*F+F(*&F)F+F'F+F(F+F6F+F+*&,2FPF+FQF+FRF+*&FJF+F*F+F(\"\"%F(*&FMF+FDF+F(FNF+F@F(F+-F-6$F',&F)F+F*F+F+F+,**&,>F)F+FNF(*(FVF+F'F+FAF+F+*(FVF+FOF+FDF+F(*&\"\"'F+FAF+F+*$)F'FVF+F+*&\"\"(F+F'F+F+*&\"\"*F+FOF+F+*&F[oF+FDF+F+F*F+*&FMF+)F'FBF+F+*(F)F+F'F+FDF+F(*&FdoF+F*F+F+*(FBF+F'F+F*F+F(F+F2F+F+*&,4*(\"#:F+F'F+F*F+F(*&\"#RF+F'F+F(*(F)F+FdoF+F*F+F(*&FaoF+F*F+F(*&F)F+F]oF+F(*&\"#LF+FOF+F(\"#=F(*&\"#8F+FdoF+F(*(FaoF+FOF+F*F+F(F+F,F+F+*&,0*(FVF+FOF+F*F+F+FhnF(FjnF(FboF(*(F[oF+F'F+F*F+F+FinF+FeoF+F+F6F+F+*&,4*&\"#KF+F'F+F+F\\oF+*&FJF+FdoF+F+*&\"#CF+FOF+F+\"#;F+FUF+FfoF+*(F[oF+FOF+F*F+F+*(\"#7F+F'F+F*F+F+F+-F-6$,&F'F+F)F+F*F+F+" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 29 "and similarly we multiply by " }{XPPEDIT 18 0 "c[n,k];" "6#&%\"cG6$%\"nG%\"kG" }{TEXT 208 1 ":" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 77 "Mgfun[`sys*sys`](%,Mgfun[dfinite_expr_to_sys](c[n,k],_f(n::shift,k::shift)));" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#<%,**&,bp*(\"#6\"\"\")%\"nG\"\"#F)%\"kGF)!\"\"*&\"#bF))F-\"\"&F)F.*(\"\"*F))F+\"\"$F)F-F)F)*(F2F)F+F))F-F6F)F.*&\"#IF))F-\"\"'F)F.*&F<F))F-\"\"(F)F.*(\"#`F)F*F))F-\"\"%F)F)*(\"#TF)F*F))F-F,F)F.*(\"#XF)F1F)F+F)F.*(\"#<F))F+F<F)F-F)F)*(F<F))F+F2F)F8F)F.*(F,F)FKF)FFF)F.*(\"#?F)FMF)FFF)F.*(\"#;F)F8F))F+FCF)F.*&FCF)F+F)F)*&FCF)F-F)F.*(F6F))F+F?F)F-F)F)*&\"#DF)F8F)F.*$)F+\"\")F)F)*&F4F)FWF)F)*&\"#&)F)FSF)F)*&\"#hF)F5F)F)*&\"#CF)F*F)F)*&\"#7F)FFF)F.*&\"#[F)FBF)F.*(F_oF)F+F)F-F)F.*(FYF)F+F)FFF)F.*(\"#PF)F8F)F*F)F)*(\"#>F)FBF)F+F)F.*(\"#nF)FFF)F5F)F.*(F2F)F*F)F1F)F)*(FCF)F*F)F;F)F.*(FeoF)F5F)FBF)F)*(F?F)F5F)F1F)F)*(F2F)FSF)FBF)F)*(\"#EF)F;F)F+F)F.*(FCF)F>F)F+F)F.*(\"#MF)FSF)F-F)F)*(\"#dF)FSF)FFF)F.*(\"#OF)FMF)F-F)F)*&FcpF)FKF)F)*&\"#qF)FMF)F)*(F4F)F5F)F8F)F)F)-%#_fG6$F+F-F)F)*&,ho*(\"$Z%F)F*F)F-F)F)*&F4F)F1F)F)*(\"$4&F)F5F)F-F)F)*(\"$<\"F)F+F)F8F)F.*(F6F)F*F)FBF)F)*(FHF)F*F)FFF)F.*(\"#:F)F1F)F+F)F)*(FYF)FKF)F-F)F)*(FCF)FMF)F8F)F.*(FCF)FKF)FFF)F)*(F`pF)FMF)FFF)F)*(FcpF)F8F)FSF)F.*&FgpF)F+F)F.*&FgpF)F-F)F)*(F,F)FWF)F-F)F)*&\"#FF)F8F)F.*&F,F)FenF)F.*&\"#@F)FWF)F.*&\"$l$F)FSF)F.*&\"$L$F)F5F)F.*&\"$o\"F)F*F)F.*&FgpF)FFF)F.*&\"#=F)FBF)F)*(\"$/#F)F+F)F-F)F)*(\"#()F)F+F)FFF)F.*(\"$l\"F)F8F)F*F)F.*(FhrF)FBF)F+F)F)*(\"#VF)FFF)F5F)F)*(F4F)F*F)F1F)F)*(F2F)F5F)FBF)F.*(F,F)F5F)F1F)F)*(F,F)FSF)FBF)F.*(\"$K$F)FSF)F-F)F)*(\"#gF)FSF)FFF)F)*(\"$D\"F)FMF)F-F)F)*&\"#&*F)FKF)F.*&\"$S#F)FMF)F.*(\"$3\"F)F5F)F8F)F.F)-F]q6$,&F+F)F)F)F-F)F)*&,J*(\"$!=F)FBF)F+F)F)*&F_oF)F-F)F)*(\"#SF)F*F)FBF)F)*(\"$5\"F)F1F)F+F)F)*(\"#uF)F+F)FFF)F)*(FduF)F8F)F*F)F)*(FPF)F*F)FFF)F)*&\"#UF)F;F)F)*&\"#mF)FFF)F)*(FPF)F*F)F1F)F)FapF)F[pF)*&FauF)FBF)F)*&\"$]\"F)F8F)F)F=F)*&\"$?\"F)F1F)F)*(\"$g\"F)F+F)F8F)F)*(FCF)F*F)F-F)F)*(FcpF)F;F)F+F)F)*(\"#9F)F+F)F-F)F)F)-F]q6$F+,&F-F)F)F)F)F)*&,ho*(\"$C'F)F*F)F-F)F.*&FfnF)F1F)F.*(\"$3)F)F5F)F-F)F.*(\"$[$F)F+F)F8F)F.*(\"#'*F)F*F)FBF)F)*(\"$G)F)F*F)FFF)F)*(F_oF)F1F)F+F)F.*(\"#aF)FKF)F-F)F.*(\"#5F)FMF)F8F)F.*(F_xF)FKF)FFF)F)*(FhwF)FMF)FFF)F)*(\"#%)F)F8F)FSF)F.*&\"#KF)F+F)F)*&FexF)F-F)F.*(F2F)FWF)F-F)F.*&\"$/\"F)F8F)F.FZF)*&F_oF)FWF)F)*&\"$!GF)FSF)F)*&\"$s#F)F5F)F)*&\"$W\"F)F*F)F)*&FhwF)FFF)F)F`oF)*(FhtF)F+F)F-F)F.*(\"$c%F)F+F)FFF)F)*(\"$Y%F)F8F)F*F)F.*(\"$7\"F)FBF)F+F)F)*(\"$e(F)FFF)F5F)F)*(F<F)F*F)F1F)F.*(FgpF)F5F)FBF)F)*&F5F)F1F)F.F^pF)*(\"$'eF)FSF)F-F)F.*(\"$v$F)FSF)FFF)F)*(\"$V#F)FMF)F-F)F.*&F[oF)FKF)F)*&\"$q\"F)FMF)F)*(\"$x#F)F5F)F8F)F.F)-F]q6$,&F+F)F,F)F-F)F),**&,B*&F,F)FFF)F)*&F2F)F8F)F)*&F2F)F5F)F)*&FCF)FBF)F)*&F,F)F*F)F)*(F2F)F+F)FFF)F.*(F2F)F*F)F-F)F.*&FCF)FSF)F)*(FCF)F+F)F-F)F.*&FBF)F+F)F)*$FMF)F)*(F,F)F8F)F*F)F.*&FSF)F-F)F)*(F,F)FFF)F5F)F.*(FfnF)F*F)FFF)F.*$F1F)F)F)F\\qF)F)*&,JF\\\\lF)*&F<F)FSF)F.*(FRF)F+F)F-F)F)*&\"#8F)F5F)F.*&F_oF)F*F)F.F[\\lF)*(F6F)F+F)FFF)F.FTF.F^\\lF)Fj[lF.*&FCF)FFF)F.Fh[lF.*(FfnF)F+F)F8F)F.FUF)*&F,F)FBF)F)*(FCF)F*F)FFF)F)*&F6F)F8F)F.*(FgoF)F*F)F-F)F)Fi[lF.*(FfnF)F5F)F-F)F)F)F[uF)F)*&,8*&FjvF)F8F)F.F+F.*(FCF)F+F)F8F)F.F,F.F^\\lF.*&FRF)FFF)F.Fh[lF.Fg[lF.*&F4F)F-F)F.*&F<F)FBF)F.*(F<F)F+F)FFF)F.F)F[wF)F)*&,6F+F)Fg\\lF.F-F.*&F<F)F8F)F.Fa]lF)Fh[lF)F^\\lF.Fb[lF.Fg[lF)Fe]lF)F)-F]q6$F]uF]wF)F),(*&,N*&F*F)F-F)F)*&F2F)F1F)F)*(F_xF)F5F)F-F)F)*(F_xF)F+F)F8F)F.*$F;F)F)FhqF.F]\\lF.*&F?F)F8F)F)*$FSF)F.*&F6F)F5F)F)Fc[lF)F_[lF)*&F4F)FBF)F)Fg[lF.*(F(F)F+F)FFF)F.*(F_xF)F8F)F*F)F.*(F6F)FBF)F+F)F.*(F<F)FFF)F5F)F)*(F2F)FSF)F-F)F)*(F6F)FSF)FFF)F)*$FKF)F.*&F6F)FMF)F.F)F\\qF)F)*&,P*(F6F)FSF)F-F)F.*(F<F)F5F)F-F)F.*(\"#GF)F+F)F-F)F)*&FRF)F1F)F.*(FYF)F*F)F-F)F)*(FexF)F+F)FFF)F)F_xF.*(FRF)F+F)F8F)F)*(\"#JF)F*F)FFF)F)*(FRF)F8F)F*F)F)FhqF)Fi^lF)*&FSF)FFF)F.F\\\\lF.*&F?F)F*F)F)*&F4F)F+F)F)*&\"#\"*F)FFF)F.*&\"#ZF)F-F)F.*&\"#$*F)F8F)F.*&FCF)F5F)F.*&F,F)F;F)F.*&F,F)FSF)F.*&FAF)FBF)F.F)F[wF)F)*&,0*&FfvF)FFF)F)*&FhyF)F-F)F)*&FdvF)F8F)F)*&\"#]F)FBF)F)*&F(F)F1F)F)Fb^lF)FexF)F)-F]q6$F+,&F,F)F-F)F)F)" }}
}
}
{SECT 0 {PARA 209 "" 0 "" {TEXT 214 20 "Creative telescoping" }}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 92 "This is the time-consuming part of this worksheet: we now compute a recurrence satisfied by " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 94 " by applying creative telescoping to the summand defined by the system we have just computed. " }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 91 "The following instruction triggers the display of extra information during the computation:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 20 "infolevel[Mgfun]:=5:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 62 "res:=Mgfun[creative_telescoping](LFSol(%%),n::shift,k::shift);" }}
{PARA 211 "" 1 "" {TEXT -1 47 "Mgfun/chyzak97:   \"Suitable term order guessed\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "Mgfun/chyzak97:   \"Implicit d-finite expression recognized\"" }}
{PARA 211 "" 1 "" {TEXT -1 34 "Mgfun/chyzak97:   \"Dimension is 3\"" }}
{PARA 211 "" 1 "" {TEXT -1 47 "Mgfun/chyzak97:   \"Start of Chyzak's algorithm\"" }}
{PARA 211 "" 1 "" {TEXT -1 62 "Mgfun/chyzak97:   \"Preparation of the system: .60e-1 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 50 "Mgfun/uncoupling:   \"Uncoupling of the LOF system\"" }}
{PARA 211 "" 1 "" {TEXT -1 60 "Mgfun/chyzak97:   \"Uncoupling of the system: 1.011 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 69 "Mgfun/rational_sys_solve:   \"Look for rational solutions of a system\"" }}
{PARA 211 "" 1 "" {TEXT -1 79 "`Mgfun/denominator_bound`[shift]   \"Intermediate bound on dispersion: infinity\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "`Mgfun/denominator_bound`[shift]   \"Bound on dispersion: 0\"" }}
{PARA 211 "" 1 "" {TEXT -1 75 "`Mgfun/denominator_bound`[shift]   \"Computing a resultant: .11e-1 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 78 "`Mgfun/rational_solve`[shift]   \"Computing denominator bound: .20e-1 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 1\"" }}
{PARA 211 "" 1 "" {TEXT -1 61 "Mgfun/rational_sys_solve:   \"Solving equation: .580 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 2\"" }}
{PARA 211 "" 1 "" {TEXT -1 63 "Mgfun/rational_sys_solve:   \"Solving equation: .10e-1 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 3\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "Mgfun/rational_sys_solve:   \"Solving equation: 0. seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 42 "Mgfun/chyzak97:   \"No LOFE of order 0 :-(\"" }}
{PARA 211 "" 1 "" {TEXT -1 62 "Mgfun/chyzak97:   \"Preparation of the system: .70e-1 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 50 "Mgfun/uncoupling:   \"Uncoupling of the LOF system\"" }}
{PARA 211 "" 1 "" {TEXT -1 60 "Mgfun/chyzak97:   \"Uncoupling of the system: 1.160 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 69 "Mgfun/rational_sys_solve:   \"Look for rational solutions of a system\"" }}
{PARA 211 "" 1 "" {TEXT -1 79 "`Mgfun/denominator_bound`[shift]   \"Intermediate bound on dispersion: infinity\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "`Mgfun/denominator_bound`[shift]   \"Bound on dispersion: 0\"" }}
{PARA 211 "" 1 "" {TEXT -1 75 "`Mgfun/denominator_bound`[shift]   \"Computing a resultant: .19e-1 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 78 "`Mgfun/rational_solve`[shift]   \"Computing denominator bound: .39e-1 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 1\"" }}
{PARA 211 "" 1 "" {TEXT -1 62 "Mgfun/rational_sys_solve:   \"Solving equation: 1.210 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 2\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "Mgfun/rational_sys_solve:   \"Solving equation: 0. seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 3\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "Mgfun/rational_sys_solve:   \"Solving equation: 0. seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 42 "Mgfun/chyzak97:   \"No LOFE of order 1 :-(\"" }}
{PARA 211 "" 1 "" {TEXT -1 60 "Mgfun/chyzak97:   \"Preparation of the system: .350 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 50 "Mgfun/uncoupling:   \"Uncoupling of the LOF system\"" }}
{PARA 211 "" 1 "" {TEXT -1 60 "Mgfun/chyzak97:   \"Uncoupling of the system: 3.360 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 69 "Mgfun/rational_sys_solve:   \"Look for rational solutions of a system\"" }}
{PARA 211 "" 1 "" {TEXT -1 79 "`Mgfun/denominator_bound`[shift]   \"Intermediate bound on dispersion: infinity\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "`Mgfun/denominator_bound`[shift]   \"Bound on dispersion: 0\"" }}
{PARA 211 "" 1 "" {TEXT -1 75 "`Mgfun/denominator_bound`[shift]   \"Computing a resultant: .60e-1 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 76 "`Mgfun/rational_solve`[shift]   \"Computing denominator bound: .119 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 1\"" }}
{PARA 211 "" 1 "" {TEXT -1 62 "Mgfun/rational_sys_solve:   \"Solving equation: 9.260 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 2\"" }}
{PARA 211 "" 1 "" {TEXT -1 63 "Mgfun/rational_sys_solve:   \"Solving equation: .10e-1 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 3\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "Mgfun/rational_sys_solve:   \"Solving equation: 0. seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 42 "Mgfun/chyzak97:   \"No LOFE of order 2 :-(\"" }}
{PARA 211 "" 1 "" {TEXT -1 60 "Mgfun/chyzak97:   \"Preparation of the system: .869 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 50 "Mgfun/uncoupling:   \"Uncoupling of the LOF system\"" }}
{PARA 211 "" 1 "" {TEXT -1 60 "Mgfun/chyzak97:   \"Uncoupling of the system: 2.831 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 69 "Mgfun/rational_sys_solve:   \"Look for rational solutions of a system\"" }}
{PARA 211 "" 1 "" {TEXT -1 79 "`Mgfun/denominator_bound`[shift]   \"Intermediate bound on dispersion: infinity\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "`Mgfun/denominator_bound`[shift]   \"Bound on dispersion: 0\"" }}
{PARA 211 "" 1 "" {TEXT -1 75 "`Mgfun/denominator_bound`[shift]   \"Computing a resultant: .11e-1 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 78 "`Mgfun/rational_solve`[shift]   \"Computing denominator bound: .11e-1 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 1\"" }}
{PARA 211 "" 1 "" {TEXT -1 63 "Mgfun/rational_sys_solve:   \"Solving equation: 46.491 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 79 "`Mgfun/denominator_bound`[shift]   \"Intermediate bound on dispersion: infinity\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "`Mgfun/denominator_bound`[shift]   \"Bound on dispersion: 0\"" }}
{PARA 211 "" 1 "" {TEXT -1 73 "`Mgfun/denominator_bound`[shift]   \"Computing a resultant: .401 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 76 "`Mgfun/rational_solve`[shift]   \"Computing denominator bound: .511 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 2\"" }}
{PARA 211 "" 1 "" {TEXT -1 62 "Mgfun/rational_sys_solve:   \"Solving equation: 4.259 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 3\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "Mgfun/rational_sys_solve:   \"Solving equation: 0. seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 42 "Mgfun/chyzak97:   \"No LOFE of order 3 :-(\"" }}
{PARA 211 "" 1 "" {TEXT -1 61 "Mgfun/chyzak97:   \"Preparation of the system: 1.980 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 50 "Mgfun/uncoupling:   \"Uncoupling of the LOF system\"" }}
{PARA 211 "" 1 "" {TEXT -1 60 "Mgfun/chyzak97:   \"Uncoupling of the system: 6.250 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 69 "Mgfun/rational_sys_solve:   \"Look for rational solutions of a system\"" }}
{PARA 211 "" 1 "" {TEXT -1 79 "`Mgfun/denominator_bound`[shift]   \"Intermediate bound on dispersion: infinity\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "`Mgfun/denominator_bound`[shift]   \"Bound on dispersion: 0\"" }}
{PARA 211 "" 1 "" {TEXT -1 73 "`Mgfun/denominator_bound`[shift]   \"Computing a resultant: .120 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 76 "`Mgfun/rational_solve`[shift]   \"Computing denominator bound: .131 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 1\"" }}
{PARA 211 "" 1 "" {TEXT -1 64 "Mgfun/rational_sys_solve:   \"Solving equation: 584.509 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 79 "`Mgfun/denominator_bound`[shift]   \"Intermediate bound on dispersion: infinity\"" }}
{PARA 211 "" 1 "" {TEXT -1 59 "`Mgfun/denominator_bound`[shift]   \"Bound on dispersion: 0\"" }}
{PARA 211 "" 1 "" {TEXT -1 73 "`Mgfun/denominator_bound`[shift]   \"Computing a resultant: .609 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 76 "`Mgfun/rational_solve`[shift]   \"Computing denominator bound: .900 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 2\"" }}
{PARA 211 "" 1 "" {TEXT -1 65 "Mgfun/rational_sys_solve:   \"Solving equation: 5801.681 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 49 "Mgfun/rational_sys_solve:   \"Equations solved: 3\"" }}
{PARA 211 "" 1 "" {TEXT -1 64 "Mgfun/rational_sys_solve:   \"Solving equation: 479.559 seconds.\"" }}
{PARA 211 "" 1 "" {TEXT -1 42 "Mgfun/chyzak97:   \"LOFE of order 4 found!\"" }}
{PARA 207 "" 1 "" {XPPMATH 20 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}}
}
}
{SECT 0 {PARA 209 "" 1 "" {XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 214 34 " satisfies the same recurrence as " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 14 "As we did for " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 208 67 ", we now sum this for k from 0 to n+4 and we observe the following:" }}
}
{SECT 0 {PARA 210 "" 0 "" {TEXT 215 35 "The right-hand side telescopes to 0" }}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 10 "R:=res[2]:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 12 "eval(R,k=0);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 49 "F:=c[n,k]*(Sum(1/m^3,m=1..n)+Sum(d[n,m],m=1..k));" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#>%\"FG*()-%)binomialG6$%\"nG%\"kG\"\"#\"\"\")-F(6$,&F*F-F+F-F+F,F-,&-%$SumG6$*&F-F-*$)%\"mG\"\"$F-!\"\"/F9;F-F*F--F46$,$*&#F-F,F-**)F;,&F9F-F-F-F-F9!\"$-F(6$F*F9F;-F(6$,&F*F-F9F-F9F;F-F-/F9;F-F+F-F-" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 45 "ff:=eval(R,_f=proc(N,K) subs(n=N,k=K,F) end):" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 23 "expand(eval(ff,k=n+5));" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 68 "Thus, we have found the following 4th order recurrence satisfied by " }{XPPEDIT 18 0 "b[n] " "6#&%\"bG6#%\"nG" }{TEXT 208 1 ":" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 54 "eval(collect(res[1],_f,factor),_f=proc(n,k) A(n) end);" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#,,*(,&*&\"\"#\"\"\"%\"nGF(F(\"\"&F(F(,8*&\"&'*Q\"F()F)\"#5F(F(*&\"'+uMF()F)\"\"*F(F(*&\"()**oQF()F)\"\")F(F(*&\")g*p_#F()F)\"\"(F(F(*&\"*C(fr5F()F)\"\"'F(F(*&\"*g$*>3$F()F)F*F(F(*&\"*88o3'F()F)\"\"%F(F(*&\"*Ic$\\\")F()F)\"\"$F(F(*&\"*xdy2(F()F)F'F(F(*&\"*5N3g$F(F)F(F(\")3_\\\")F(F(-%\"AG6#,&F)F(F'F(F(F(**,&F&F(FJF(F(,&F&F(F;F(F(,6*&\"$3%F(F2F(F(*&\"%czF(F6F(F(*&\"&'3oF(F:F(F(*&\"'%GO$F(F>F(F(*&\"(!*)e5F(FBF(F(*&\"(n(4AF(FEF(F(*&\"(1K1$F(FIF(F(*&\"(*yCFF(FMF(F(*&\"(1IT\"F(F)F(F(\"'kcKF(F(-FR6#,&F)F(F(F(F(!\"\"**FVF(,,*&\"#7F(FEF(F(*&\"#'*F(FIF(F(*&\"$$GF(FMF(F(*&\"$k$F(F)F(F(\"$t\"F(F(),&F)F(FFF(F?F(-FR6#FfpF(F(**FWF(,,F\\pF(*&\"$W\"F(FIF(F(*&\"$V'F(FMF(F(*&\"%m7F(F)F(F(\"$G*F(F()FhoF?F(-FR6#F)F(F(**FVF(FWF(,6FYF(*&\"&//\"F(F6F(F(*&\"'Yq6F(F:F(F(*&\"'E:wF(F>F(F(*&\"(?N:$F(FBF(F(*&\"(Lsg)F(FEF(F(*&\");;Y:F(FIF(F(*&\"),?g<F(FMF(F(*&\")m&4:\"F(F)F(F(\"(;5H$F(F(-FR6#,&F)F(FJF(F(Fio" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 8 "rec2:=%:" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 7 "Check: " }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 56 "seq(eval(subs(n=i,rec2),A=proc(n) listb[n] end),i=1..6);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6(\"\"!F#F#F#F#F#" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 22 "Now, we observe using " }{HYPERLNK 211 "gfun" 2 "gfun" "" }{TEXT 208 87 "  that all solutions of the second order recurrence rec found above to be satisfied by " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 208 39 " are also solutions of this recurrence:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 44 "expand(gfun[`rec+rec`](rec,rec2,A(n))-rec2);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 95 "This shows that rec2 is also satisfied by all linear combinations of solutions of rec and rec2." }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 25 "Therefore, to prove that " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 105 " satisfies rec, it is sufficient to check that its first 4 values satisfy it, which has been done above. " }}
}
}
}
}
{SECT 0 {PARA 206 "" 1 "" {TEXT 210 27 "4. Asymptotic behaviour of " }{XPPEDIT 18 0 "c[n];" "6#&%\"cG6#%\"nG" }{TEXT 210 2 ":=" }{XPPEDIT 18 0 "a[n]*zeta(3)-b[n];" "6#,&*&&%\"aG6#%\"nG\"\"\"-%%zetaG6#\"\"$F)F)&%\"bGF'!\"\"" }}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 42 "We start from the recurrence satisfied by " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 208 2 ", " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 5 " and " }{XPPEDIT 18 0 "c[n];" "6#&%\"cG6#%\"nG" }{TEXT 208 38 " defined in the title of this section:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 4 "rec;" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#,(*&),&%\"nG\"\"\"F(F(\"\"$F(-%\"AG6#F'F(!\"\"*&),&F'F(\"\"#F(F)F(-F+6#F0F(F-*(,&*&F1F(F'F(F(F)F(F(,(*&\"#<F()F'F1F(F(*&\"#^F(F'F(F(\"#RF(F(-F+6#F&F(F(" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 89 "and consider the differential equation satisfied by generating functions of its solutions" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 33 "gfun[rectodiffeq](rec,A(n),y(z));" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#<%/-%\"yG6#\"\"!&%#_CGF'/--%\"DG6#F&F'&F*6#\"\"\",.*&,&%\"zG!\"\"\"\"&F2F2-F&6#F6F2F2*&,(*&\"\"(F2)F6\"\"#F2F7*&\"$7\"F2F6F2F2F2F7F2-%%diffG6$F9F6F2F2*&,(*&\"\"'F2)F6\"\"$F2F7*&\"$`\"F2F?F2F2*&FKF2F6F2F7F2-FD6$F9-%\"$G6$F6F@F2F2*&,(*$)F6\"\"%F2F7*&\"#MF2FJF2F2*$F?F2F7F2-FD6$F9-FR6$F6FKF2F2*&F8F2F)F2F7F0F2" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 25 "we make this homogeneous:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 95 "deq:=collect(gfun[diffeqtohomdiffeq](op(remove(type,%,`=`)),y(z))/(5*_C[0]-_C[1]),diff,factor);" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#>%$deqG,,*&,(*&\"#D\"\"\")%\"zG\"\"#F*F**&\"$=%F*F,F*!\"\"\"\"%F*F*-%%diffG6$-%\"yG6#F,-%\"$G6$F,F-F*F***\"\"&F*,(*&F-F*F+F*F**&\"#^F*F,F*F0F*F*F*F,F*-F36$F5-F96$F,\"\"$F*F**&,&*&\"#:F*F,F*F*\"$<\"F0F*-F36$F5F,F*F**(,(*$F+F*F**&\"#MF*F,F*F0F*F*F*F+F*-F36$F5-F96$F,F1F*F*F5F*" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 70 "The solutions of this equation (and therefore the generating functions" }{XPPEDIT 18 0 "A(z);" "6#-%\"AG6#%\"zG" }{TEXT 208 2 ", " }{XPPEDIT 18 0 "B(z);" "6#-%\"BG6#%\"zG" }{TEXT 208 5 " and " }{XPPEDIT 18 0 "C(z);" "6#-%\"CG6#%\"zG" }{TEXT 208 18 " of the sequences " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 208 2 ", " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 5 " and " }{XPPEDIT 18 0 "c[n];" "6#&%\"cG6#%\"nG" }{TEXT 208 36 ") may have singularities only at 0, " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT 208 67 ", and any of the roots of the leading coefficient of this equation:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 41 "polsing:=coeff(deq,diff(y(z),[z$4]))/z^2;" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#>%(polsingG,(*$)%\"zG\"\"#\"\"\"F**&\"#MF*F(F*!\"\"F*F*" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 37 "Here are the roots of this polynomial" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 17 "solve(polsing,z);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6$,&\"#<\"\"\"*&\"#7F%\"\"##F%F(F%,&F$F%F&!\"\"" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 38 "and the corresponding numerical values" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 11 "evalf([%]);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#7$$\"+ui0(R$!\")$\"(EP%HF&" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 256 "First, it should be observed (by simple bounds on the binomial coefficients) that the sequences do not grow so fast as to make their generating series divergent. Therefore 0 is not a singularity of the functions we're interested in. Secondly, the sequence " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 208 43 " is clearly increasing, which implies that " }{XPPEDIT 18 0 "A(z);" "6#-%\"AG6#%\"zG" }{TEXT 208 62 " has a singularity of modulus smaller than 1, which has to be " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 208 1 "=" }{XPPEDIT 18 0 "17-12*sqrt(2);" "6#,&\"#<\"\"\"*&\"#7F%-%%sqrtG6#\"\"#F%!\"\"" }{TEXT 208 8 ". Since " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 1 "/" }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 208 51 " has a finite limit, this is also a singularity of " }{XPPEDIT 18 0 "B(z);" "6#-%\"BG6#%\"zG" }{TEXT 208 49 ". A crucial step of the proof is to observe that " }{XPPEDIT 18 0 "C(z);" "6#-%\"CG6#%\"zG" }{TEXT 208 4 " is " }{TEXT 213 3 "not" }{TEXT 208 79 " singular at that point and therefore its singularity of smallest modulus is 1/" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 208 1 "=" }{XPPEDIT 18 0 "17+12*sqrt(2);" "6#,&\"#<\"\"\"*&\"#7F%-%%sqrtG6#\"\"#F%F%" }{TEXT 208 34 ", whence an exponential growth of " }{XPPEDIT 18 0 "c[n];" "6#&%\"cG6#%\"nG" }{TEXT 208 5 "  in " }{XPPEDIT 18 0 "alpha^n;" "6#)%&alphaG%\"nG" }{TEXT 208 108 ". This step is achieved by considering the local behaviour of the solutions of the differential equation at " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 208 1 ":" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 77 "alias(alpha=RootOf(polsing)):DEtools[formal_sol](deq,y(z),T,z=alpha,terms=4);" }}
{PARA 207 "" 1 "" {XPPMATH 20 "6#7$7&+-%\"TG#\"\"\"\"\"#\"\"!,&#\"&>T&\"%!)G!\"\"*(\"%\"f\"F(F.F/%&alphaGF(F(F(,&#\"'\"RQ\"\"$C#F(*(\"&*=EF(\"%S9F/F2F(F/F),&#\"+\\miq#)\"'+OXF/*(\"*L!Hp[F(\"'+s!*F/F2F(F(\"\"$-%\"OG6#F(\"\"%++F&F/F(,&#F-F9F(*(F1F(F9F/F2F(F/F),&#\"&&GT\"#OF/*(\"&2q#F(\"$+)F/F2F(F(FAFBFE+)F&F(F),&FHF/FIF(FAFBFE/F&,&%\"zGF(F2F/7$*&F&F'+-F&F(F*,&#\"%2p\"$%QF/*(\"$.#F(FfnF/F2F(F(F(,&#\"(dhI\"\"%sIF(*(\"&JS#F(\"%?>F/F2F(F/F),&#\"-*p//di%\")!o(GTF/*(\",B8!oh8F(FcoF/F2F(F(FAFBFEF(FS" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 52 "The interpretation of this result is as follows: at " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 208 211 ", the differential equation has a vector space of dimension 4 of solutions, which is the direct sum of a subspace of analytic solutions of dimension 3 and a \"line\" of singular solutions, generated by a solution " }{XPPEDIT 18 0 "L(z) = sqrt(alpha-z)*(1+O(z-alpha));" "6#/-%\"LG6#%\"zG*&-%%sqrtG6#,&%&alphaG\"\"\"F'!\"\"F.,&F.F.-%\"OG6#,&F'F.F-F/F.F." }{TEXT 208 7 ". Thus " }{XPPEDIT 18 0 "A(z) = lambda[A]*L(z)+g[A](z);" "6#/-%\"AG6#%\"zG,&*&&%'lambdaG6#F%\"\"\"-%\"LGF&F-F--&%\"gGF,F&F-" }{TEXT 208 8 ", where " }{XPPEDIT 18 0 "g[A](z);" "6#-&%\"gG6#%\"AG6#%\"zG" }{TEXT 208 36 " is analytic in the neighborhood of " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 208 15 " and similarly " }{XPPEDIT 18 0 "B(z) = lambda[B]*L(z)+g[B](z);" "6#/-%\"BG6#%\"zG,&*&&%'lambdaG6#F%\"\"\"-%\"LGF&F-F--&%\"gGF,F&F-" }{TEXT 208 59 ". Now, singularity analysis tells us that the coefficients " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 208 27 " behave asymptotically like" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 40 "equivalent(lambda[A]*sqrt(alpha-z),z,n);" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&*,%#PiG#!\"\"F'&%'lambdaG6#%\"AGF&%&alphaGF%*&F&F&%\"nGF+#\"\"$F'),&\"#MF&F0F+,$-%#lnG6#-%$expG6#,$F2F+F+F&F&F+-%\"OG6#*&)F1#\"\"&F'F&)F6F2F&F&" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 18 "and similarly for " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 6 " with " }{XPPEDIT 18 0 "lambda[B];" "6#&%'lambdaG6#%\"BG" }{TEXT 208 13 " in place of " }{XPPEDIT 18 0 "lambda[A];" "6#&%'lambdaG6#%\"AG" }{TEXT 208 26 ". Now, we deduce that lim(" }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT 208 1 "/" }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT 208 2 ")=" }{XPPEDIT 18 0 "zeta(3);" "6#-%%zetaG6#\"\"$" }{TEXT 208 1 "=" }{XPPEDIT 18 0 "lambda[B];" "6#&%'lambdaG6#%\"BG" }{TEXT 208 1 "/" }{XPPEDIT 18 0 "lambda[A];" "6#&%'lambdaG6#%\"AG" }{TEXT 208 18 " and consequently " }{XPPEDIT 18 0 "C(z);" "6#-%\"CG6#%\"zG" }{TEXT 208 1 "=" }{XPPEDIT 18 0 "A(z)*zeta(3)-B(z);" "6#,&*&-%\"AG6#%\"zG\"\"\"-%%zetaG6#\"\"$F)F)-%\"BGF'!\"\"" }{TEXT 208 16 " is analytic at " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 208 22 ", as was to be proved." }}
}
}
{SECT 0 {PARA 206 "" 1 "" {TEXT 210 3 "5. " }{XPPEDIT 18 0 "0 < zeta(3)-b[n]/a[n];" "6#2\"\"!,&-%%zetaG6#\"\"$\"\"\"*&&%\"bG6#%\"nGF*&%\"aGF.!\"\"F2" }}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 69 "The proof is obtained by writing this difference as the infinite sum " }{XPPEDIT 18 0 "Sum(b[k]/a[k]-b[k-1]/a[k-1],k = n+1 .. infinity)" "6#-%$SumG6$,&*&&%\"bG6#%\"kG\"\"\"&%\"aGF*!\"\"F,*&&F)6#,&F+F,F,F/F,&F.F2F/F//F+;,&%\"nGF,F,F,%)infinityG" }}
}
{EXCHG {PARA 204 "" 0 "" {TEXT 208 83 "A recurrence satisfied by the numerator of the summand is readily found using gfun:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 91 "gfun[poltorec](A(k)*B(k+1)-A(k+1)*B(k),[subs(n=k,rec),subs(A=B,n=k,rec)],[A(k),B(k)],u(k));" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#,&*&,**$)%\"kG\"\"$\"\"\"!\"\"*&F)F*)F(\"\"#F*F+*&F)F*F(F*F+F*F+F*-%\"uG6#F(F*F**&,*\"\")F**&\"#7F*F(F*F**&\"\"'F*F-F*F*F&F*F*-F16#,&F(F*F*F*F*F*" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 15 "rsolve(%,u(k));" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#*&,&%\"kG\"\"\"F&F&!\"$-%\"uG6#\"\"!F&" }}
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 47 "and thus this is a sum of positive terms since " }{XPPEDIT 18 0 "0 < a[k];" "6#2\"\"!&%\"aG6#%\"kG" }{TEXT 208 39 " and the initial condition is positive:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 67 "value(subs(n=0,a[n])*subs(n=1,b[n])-subs(n=1,a[n])*subs(n=0,b[n]));" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#\"\"'" }}
}
}
{SECT 0 {PARA 206 "" 0 "" {TEXT 210 13 "6. Conclusion" }}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 34 "It is classical that the sequence " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#%\"nG" }{TEXT 208 11 "=lcm(1,...," }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 208 28 ") grows asymptotically like " }{XPPEDIT 18 0 "e^n;" "6#)%\"eG%\"nG" }{TEXT 208 51 ". This worksheet has thus proved that the sequence " }{XPPEDIT 18 0 "S[n];" "6#&%\"SG6#%\"nG" }{TEXT 208 1 "=" }{XPPEDIT 18 0 "d[n]^3*(a[n]*zeta(3)-b[n]);" "6#*&&%\"dG6#%\"nG\"\"$,&*&&%\"aGF&\"\"\"-%%zetaG6#F(F-F-&%\"bGF&!\"\"F-" }{TEXT 208 79 " enjoys the following properties: it is positive and grows asymptotically like " }{XPPEDIT 18 0 "(exp(3)*(17-12*sqrt(2)))^n;" "6#)*&-%$expG6#\"\"$\"\"\",&\"#<F)*&\"#7F)-%%sqrtG6#\"\"#F)!\"\"F)%\"nG" }{TEXT 208 24 ".  Now, this tends to 0:" }}
}
{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 209 30 "evalf(exp(3)*(17-12*sqrt(2)));" }}
{PARA 205 "" 1 "" {XPPMATH 20 "6#$\"+E<j7f!#5" }}
}
{EXCHG {PARA 204 "> " 0 "" }
}
{EXCHG {PARA 204 "" 1 "" {TEXT 208 3 "If " }{XPPEDIT 18 0 "zeta(3);" "6#-%%zetaG6#\"\"$" }{TEXT 208 25 " were rational, then for " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 208 15 " large enough, " }{XPPEDIT 18 0 "d[n]^3*zeta(3);" "6#*&&%\"dG6#%\"nG\"\"$-%%zetaG6#F(\"\"\"" }{TEXT 208 38 " would be an integer and the sequence " }{XPPEDIT 18 0 "S[n];" "6#&%\"SG6#%\"nG" }{TEXT 208 154 " would be a sequence of positive integers tending to 0, which is impossible. This concludes the proof. Further refinements lead to irrationality measures." }}
}
}
{PARA 212 "" 0 "" }
}
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