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Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent. (English) Zbl 1216.11075
Summary: The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for \(\zeta (2)\) and \(\zeta (3)\), and of the second author for Euler’s constant \(\gamma\) and its alternating analog \(\ln (4/\pi)\), and on the other hand the infinite products of the first author for \(e\), of the second author for \(\pi\), and of J. Ser for \(e^\gamma\). We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch’s transcendent of Hadjicostas’s double integral formula for the Riemann zeta function, and logarithmic series for the digamma and Euler beta functions. The main tools are analytic continuations of Lerch’s function, including Hasse’s series. We also use Ramanujan’s polylogarithm formula for the sum of a particular series involving harmonic numbers, and his relations between certain dilogarithm values.

MSC:
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M35 Hurwitz and Lerch zeta functions
11Y60 Evaluation of number-theoretic constants
33B15 Gamma, beta and polygamma functions
33B30 Higher logarithm functions
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