Variations on the Sequence of Apéry Numbers

Frédéric Chyzak

(Version of January 9, 1998)

In the early 1990's, Doron Zeilberger and Herbert Wilf, developped a new methodology for symbolic summation and integration (Wilf, Herbert S. and Zeilberger, Doron (1992):An algorithmic proof theory for hypergeometric (ordinary and `` [Maple Math] '') multisum/integral identities, Inventiones Mathematicae , 108 :575-633). One of the most famous successes of this so-called ``WZ-method'' has been to provide a computer proof of the combinatorial identity

[Maple Math] ,

and to prove that the sequence of these numbers [Maple Math] satisfies the second order recurrence equation

[Maple Math] .

Proving this recurrence was a crucial step of Apéry's proof for the irrationality of

[Maple Math] .

On the other hand, the identity itself stems from a number-theoretic question raised by Schmidt in (Schmidt, Asmus L. (1990): Generalized Legendre polynomials, J. reine angew. Math. , 404 :192-202).

Several proofs of the identity above, that relates the Apéry numbers [Maple Math] to the Franel numbers

[Maple Math] ,

were given in (Strehl, Volker (1994): Binomial Identities, Combinatorial and Algorithmic Aspects, Discrete Math. , 136 :309-346). One of them in particular is based on Zeilberger's algorithm for hypergeometric summation, and yields the recurrence equation above as a by-product. In the following sections, we first recall how Apéry was led to the identity, borrowing from Van der Poorten's report (Van der Poorten, Alfred (1979): A Proof that Euler missed... Apéry's Proof of the Irrationality of [Maple Math] , Math. Intelligencer , 1 :195-203); we next give a proof for both results using our package Mgfun , and finally exploit the recurrence equation to beat Maple computing many digits of [Maple Math] .

Sketch of Apéry's Proof

Proof of Apéry's Recurrence

Computation of [Maple Math] Using Standard Maple and the Holonomic Approach