Well-poised generation of Apéry-like recursions

Details Well-poised generation of Apéry-like recursions Well-poised generation of Apéry-like recursions

2003-07-04

arxiv.org/abs/math/0307058v2

J. Comput. Appl. Math. 178:1--2 (2005), 513--521

Mathematics

Number Theory

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

Scientific paper

10.1016/j.cam.2003.11.016

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 Ap\'ery, as well as the known recursion for rational approximations to $\zeta(4)$. Multiple integral representations for solutions of the constructed recurrences are also given.

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