# A note on a generalized double series

## TL;DR

This paper uses contour integration to derive a generalized double series involving the Hurwitz-Lerch zeta function and expresses it in terms of special functions.

## Contribution

The paper introduces a new method to derive closed-form expressions for a generalized double series using contour integration.

## Key findings

- Closed-form formulae for a generalized double series are derived using the Hurwitz-Lerch zeta function.
- Special cases of summation and product formulae are expressed using trigonometric and gamma functions.
- A table and plots of quotient gamma functions are provided for reference.

## 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. A short table of quotient gamma functions and plots are produced for easy reading by interested readers.