Some families of the general Mathieu‐type series with associated properties and functional inequalities

Abstract

The main purpose of this paper is to introduce general families of the extended Mathieu‐type power series and present a number of potentially useful integral representations of several general families of the extended Mathieu‐type power series in a unified manner. Relationships of the extended Mathieu‐type functional power series with the generalized Hurwitz–Lerch zeta function is also considered. Various other properties, mainly, Mellin transform and Hankel transform, and fractional derivative formulae are derived for the extended Mathieu series. A pair of the bounding inequalities are established for the extended Mathieu‐type series. As an application of newly defined function, we present a systematic study of probability density function and distribution function associated with the general extended Mathieu‐type power series. In particular, the mathematical expectation and variance of the distribution are derived. Finally, we prove some properties of monotonicity, convexity, and Turán‐type inequalities for the general extended Mathieu‐type power series.