Reciprocal Hyperbolic Series of Ramanujan Type

TL;DR

This paper develops a method to evaluate certain infinite hyperbolic series, extending Ramanujan's work, by using contour integrals and Eisenstein series, resulting in explicit formulas involving hypergeometric and Gamma functions.

Contribution

It introduces a novel approach to summing hyperbolic series using contour integrals and special functions, providing new explicit evaluations and examples.

Findings

Several hyperbolic series are explicitly evaluated.

Series are expressed in terms of hypergeometric and Gamma functions.

New illustrative examples demonstrate the method's effectiveness.

Abstract

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of $z={}_2F_1(1/2,1/2;1;x)$ and $z'=dz/dx$. When a certain parameter in these series is equal to $\pi$ the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.