Example3-24-AperyNumbersAsDiagonals.mw

Example 3.24 (Apéry Numbers as Diagonals)

Requires BinomialSum package of Pierre Lairez, available at https://github.com/lairez/binomsums

 > # Import package with(BinomSums);

 (1)

 > ################## # zeta(3) sequence ##################

 > # The generating function for the binomial sum in Apéry's irrationality proof of zeta(3) is GF_as_sum := Sum(t^n*Sum(Binomial(n,k)^2*Binomial(n+k,k)^2, k=0..n), n=0..infinity);

 (2)

 > # The binomial sums package shows this is the "residue" of this rational function R := subs(v[1]=x,v[2]=y,v[3]=z,[sumtores(GF_as_sum, v)][1]);

 (3)

 > # This implies the GF is the main power series diagonal of this function F := simplify(x*y*z*subs(t=x*y*z*t,R));

 (4)

 > # Beginning terms in the Apéry sequence seq(coeff(coeff(coeff(coeff(mtaylor(F,[x,y,z,t],30),x,k),y,k),z,k),t,k),k=0..6);

 (5)

 > ################## # zeta(2) sequence ##################

 > # The generating function for the binomial sum in Apéry's irrationality proof of zeta(2) is GF_as_sum2 := Sum(t^n*Sum(Binomial(n,k)^2*Binomial(n+k,k), k=0..n), n=0..infinity);

 (6)

 > # The binomial sums package shows this is the "residue" of this rational function R2 := subs(v[1]=x,v[2]=y,[sumtores(GF_as_sum2, v)][1]);

 (7)

 > # This implies the GF is the main power series diagonal of this function F2 := simplify(x*y*subs(t=x*y*t,R2));

 (8)

 > # Beginning terms in the second Apéry sequence seq(coeff(coeff(coeff(mtaylor(F2,[x,y,t],20),x,k),y,k),t,k),k=0..6);

 (9)

 >