A note on a generalized double series

TL;DR

This paper employs contour integration to derive closed-form expressions for a generalized double series involving the Hurwitz-Lerch zeta function, connecting it with special functions and deriving special cases.

Contribution

It introduces a novel contour integration method to evaluate a generalized double series involving the Hurwitz-Lerch zeta function and derives related special cases.

Findings

Closed-form formulas involving special functions

Connections between series and trigonometric/gamma functions

Derivation of summation and product formulas

Abstract

By employing contour integration the derivation of a generalized double finite series involving the Hurwitz-Lerch zeta function is used to derive closed form formulae in terms of special functions. We use this procedure to find special cases of the summation and product formulae in terms of the Hurwitz-Lerch zeta function, trigonometric functions and the gamma function.