#### 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 $\zeta \left(2\right)$. 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