Vol. 9, No. 4, 2020

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Two integral transformations related to $\zeta(2)$

Raffaele Marcovecchio

Vol. 9 (2020), No. 4, 441–452
Abstract

We prove two integral transformations that relate different constructions of rational approximations to ζ(2). The first one relates a double integral over the unit square and a Barnes-type integral. The second one relates two Barnes-type integrals and was discovered and proved by W. Zudilin using an automated proof method. Here we offer a proof based on more classical means such as contiguous relations, the second Barnes lemma and the duplication formula for the gamma function.

To the memory of Naum Ilyitch Feldman (1918–1994)

Keywords
Legendre polynomials, irrationality measure, zeta values, hypergeometric functions, human-generated proofs
Mathematical Subject Classification 2010
Primary: 11J82
Secondary: 33C20, 33C60
Milestones
Received: 31 December 2019
Accepted: 5 May 2020
Published: 5 November 2020
Authors
Raffaele Marcovecchio