Recurrence for the Left-Hand Side

We first prove that the Apéry numbers, as defined by the left-hand side

[Maple Math] ,

satisfy the announced recurrence.

The summand

> [Maple Math]

satisfies both following equations:

> h(n+1,k)/h(n,k)=factor(normal(subs(n=n+1,f)/f,expanded));

[Maple Math]

> h(n,k+1)/h(n,k)=factor(normal(subs(k=k+1,f)/f,expanded));

[Maple Math]

This yields the following system

> sys:=collect(map(numer,map(eq->op(1,eq)-op(2,eq),{"","})),h);

[Maple Math]
[Maple Math]

where each element [Maple Math] in the set denotes the equation [Maple Math] . The definite summation over [Maple Math] in [Maple Math] is performed by the following call to Mgfun[sum_of_sys] :

> sum_of_sys(sys,k=-infinity..infinity,takayama_algo);

[Maple Math]

> rec[left]:=op(collect(",h,factor));

[Maple Math]

This is the announced recurrence in disguise.