__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 ``
'') 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

,

and to prove that the sequence of these numbers satisfies the second order recurrence equation

.

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

.

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 to the Franel numbers

,

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
,
*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
.

**Sketch of Apéry's Proof**

**Proof of Apéry's Recurrence**

**Computation of **
** Using Standard Maple and the Holonomic Approach**