Well-poised generation of Apéry-like recursions

Abstract

The idea to use classical hypergeometric series and, in particular, well-poised hypergeometric series in diophantine problems of the values of the polylogarithms has led to several novelties in number theory and neighbouring areas of mathematics. Here we present a systematic approach to derive second-order polynomial recursions for approximations to some values of the Lerch zeta function, depending on the fixed (but not necessarily real) parameter $\alpha$ satisfying the condition $\Re (\alpha )<1$. Substituting $\alpha =0$ into the resulting recurrence equations produces the famous recursions for rational approximations to $\zeta (2)$, $\zeta (3)$ due to Apéry, as well as the known recursion for rational approximations to $\zeta (4)$. Multiple integral representations for solutions of the constructed recurrences are also given.

8 pages; to appear in the Proceedings of the 7th OPSFA (Copenhagen, 18--22 August 2003)

References in corpus (1)

• A few remarks on linear forms involving Catalan's constant