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

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.