Recurrence for the Left-Hand Side

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

,

satisfy the announced recurrence.

The summand

>

satisfies both following equations:

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

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

This yields the following system

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

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

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

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

This is the announced recurrence in disguise.