Some Families of the Mathieu-Type Series with Certain Fractional Calculus Operators and Integral Transforms

The main motive of this paper is to create numerous new results and new findings pertaining to the Marichev-Saigo-Maeda fractional integral and fractional derivative operators applied on a new generalized family of Mathieu series, which are described in terms of the Hadamard product of the Fox-Wright function and the generalized Mathieu series. In this article, we first propose a new generalized family of functional series of the Mathieu type. Furthermore, we have also derived the fractional calculus formulae for the Saigo fractional integral and differential operators employing on generalized form of Mathieu series. Additionally, we establish a few integral transform formulas of the generalized Mathieu series. Further, some inequalities and applications of generalized Mathieu series in probability theory are studied.