Identities for Catalan's constant arising from integrals depending on a parameter

Abstract

In this paper we provide some relationships between Catalan's constant and the $_3{\rm F}_2$ and $_4{\rm F}_3$ hypergeometric functions, deriving them from some parametric integrals. In particular, using the complete elliptic integral of the first kind, we found an alternative proof of a result of Ramanujan for $_3{\rm F}_2$, a second identity related to $_4{\rm F}_3$ and using the complete elliptic integral of the second kind we obtain an identity by Adamchik.