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 ${}_3{\mathrm{F}}_2$ and ${}_4{\mathrm{F}}_3$ 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 ${}_3{\mathrm{F}}_2$, a second identity related to ${}_4{\mathrm{F}}_3$ and using the complete elliptic integral of the second kind we obtain an identity by Adamchik.

10 pages, accepted for publication