Analogues of a transformation formula of Ramanujan

TL;DR

This paper derives two new analogues of Ramanujan's transformation formula involving special functions, leading to modular relations and connections to known identities.

Contribution

It introduces novel analogues of Ramanujan's transformation formula involving Hurwitz zeta functions and modular relations, expanding the understanding of these special functions.

Findings

Derived two new analogues of Ramanujan's transformation formula.

Connected the new formulas to known identities involving polygamma functions.

Obtained Ramanujan's transformation formula as a limiting case.

Abstract

We derive two new analogues of a transformation formula of Ramanujan involving the Gamma and Riemann zeta functions present in the Lost Notebook. Both involve infinite series consisting of Hurwitz zeta functions and yield modular relations. As a special case of the first formula, we obtain an identity involving polygamma functions given by Andrew P. Guinand and as a limiting case of the second formula, we derive the transformation formula of Ramanujan.