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

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 Turan-type inequalities for the general extended Mathieu-type power series.