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

By Jesús Guillera, Jonathan Sondow. 2005

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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.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/9a4e2b019d237787c4904e5fda2a5e3bbf0fa959

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

Special functions related to Dedekind-type DC-sums and their applications (2009, methodology): “([2], [6], [16], [15], [14], [13], [24], [55], [56], [58], [29]) and see also the references cited in each of these earlier works.”
Some families of Genocchi type polynomials and their interpolation functions (2012, background): “], [2].”
Euler's constant: Euler's work and modern developments (2013, background): “In 2010 Sondow [279] gave refinements of this expansion yielding approximations with a convergence rate O( logmmr ) for an arbitrary but fixed r ≥ 2.”