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.
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