On a double series

TL;DR

This paper investigates a specific double series involving a complex summation with exponential and square root terms, aiming to establish its functional properties.

Contribution

It introduces a new analysis of a particular double series, revealing its functional characteristics and potential applications.

Findings

Derived a functional property of the double series.

Identified conditions for convergence and behavior.

Provided insights into related series and functions.

Abstract

We shall investigate and arrive at a certain functional property of the double series \[ \sum\limits_{n,r\geq 1}\frac{1}{\sqrt{x^2n^2+r^2+w^2}\left( e^{2 \pi y\sqrt{x^2n^2+r^2+w^2}}-1\right)}. \]