We prove two integral transformations that relate different constructions of rational approximations
to .
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.
