On a new (p,q)-Mtahieu type power series and its applications

TL;DR

This paper introduces new (p,q)-Mathieu power series, explores their integral representations, defines (p,q)-Mittag-Leffler functions, and establishes inequalities and relationships between these functions, expanding the theoretical framework of special functions.

Contribution

It presents novel (p,q)-Mathieu power series, their integral forms, and related (p,q)-Mittag-Leffler functions, along with new inequalities and functional relationships.

Findings

Derived Mellin-Barnes integral representations.

Introduced (p,q)-Mittag-Leffler functions and their relationships.

Proved Turán type inequalities for the (p,q)-Mathieu series.

Abstract

Our aim in this paper, is to establish certain new integrals for the the (p,q)-Mathieu--power series. In particular, we investigate the Mellin-Barnes type integral representations for a particular case of thus special function. Moreover, we introduce the notion of the (p,q)-Mittag-Leffler functions and we present a relationships between thus two functions. Some other applications are proved, in particular two Tur\'an type inequalities for the (p,q)-Mathieu series are proved.