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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.

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