The Ramanujan Journal

, Volume 16, Issue 3, pp 247–270

Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent

Authors

Article

DOI: 10.1007/s11139-007-9102-0

Cite this article as:
Guillera, J. & Sondow, J. Ramanujan J (2008) 16: 247. doi:10.1007/s11139-007-9102-0

Abstract

The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and of the second author for Euler’s constant γ and its alternating analog ln (4/π), and on the other hand the infinite products of the first author for e, of the second author for π, and of Ser for eγ. 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.

Keywords

Lerch transcendentZeta functionInfinite productDouble integralPolylogarithm

Mathematics Subject Classification (2000)

11M0611M3511Y6033B1533B30
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© Springer Science+Business Media, LLC 2008