Some infinite series involving hyperbolic functions

TL;DR

This paper introduces a contour integral method to evaluate infinite series involving hyperbolic functions, deriving explicit formulas in terms of Riemann zeta values and establishing new relations and formulas related to (2n+1).

Contribution

It presents a novel contour integral approach for hyperbolic series evaluation, providing explicit formulas and new relations, including formulas for (2n+1) similar to Ramanujan's.

Findings

Explicit formulas for hyperbolic series in terms of Riemann zeta values

New relations involving multiple hyperbolic series

Derivation of formulas for (2n+1) similar to Ramanujan's

Abstract

This paper develops an approach to the evaluation of infinite series involving hyperbolic functions. By using the approach, we give explicit formulas for several classes of series of hyperbolic functions in terms of Riemann zeta values. Moreover, we also establish many relations involving two or more series of hyperbolic functions. Furthermore, we obtain the Ramanujan's formula for \zeta(2n + 1) and find another similar formulas. The approach is based on simple contour integral representations and residue computations. Some interesting (known or new) consequences and illustrative examples are considered.