Well-poised generation of Apéry-like recursions

摘要

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 α satisfying the condition Re(α)<1. Substituting α=0 into the resulting recurrence equations produces the famous recursions for rational approximations to ζ(2), ζ(3) due to Apéry, as well as the known recursion for rational approximations to ζ(4). Multiple integral representations for solutions of the constructed recurrences are also given.