Mathieu-type series built by $(p, q)$-extended Gaussian hypergeometric function

TL;DR

This paper derives closed-form integral expressions and bounds for Mathieu-type series involving a $(p, q)$-extended Gaussian hypergeometric function, expanding the analytical tools available for these special functions.

Contribution

It introduces new integral representations and bounds for Mathieu-type series built with a $(p, q)$-extended hypergeometric function, which are novel contributions to the study of special functions.

Findings

Closed integral form expressions for Mathieu-type series.

Upper bounds for the series are established.

The results extend existing knowledge on hypergeometric functions.

Abstract

The main purpose of this paper is to present closed integral form expressions for the Mathieu-type $\boldsymbol a$-series and its associated alternating version whose terms contain a $(p, q)$-extended Gauss' hypergeometric function. Certain upper bounds for the two series are also given.