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Well-poised generation of Apéry-like recursions. (English) Zbl 1063.11026
Author’s summary: 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 a fixed (but not necessarily real) parameter \(\alpha\) satisfying \(\text{Re}(\alpha)<1\). 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.

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
33C20 Generalized hypergeometric series, \({}_pF_q\)
33B30 Higher logarithm functions
11J99 Diophantine approximation, transcendental number theory
Full Text: DOI arXiv
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