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 $α$ 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.