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)

 

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