On $p$--extended Mathieu series

TL;DR

This paper introduces new extended forms of the generalized Mathieu series, derives their integral representations, and establishes a key bounding inequality, advancing the mathematical understanding of these series.

Contribution

It presents novel extensions of Mathieu series, provides their integral representations, and proves a master bounding inequality, filling gaps in existing mathematical literature.

Findings

New extended Mathieu series introduced

Integral representations derived for the series

Master bounding inequality established

Abstract

Motivated by several generalizations of the well--known Mathieu series, the main object of this paper is to introduce new extension of generalized Mathieu series and to derive various integral representations of such series. Finally master bounding inequality is established using the newly derived integral expression.