Multi-parameter Mathieu, and alternating Mathieu series

The main purpose of this paper is to present a multi–parameter study of the familiar Mathieu series and the alternating Mathieu series S(r) and S˜(r). The computable series expansions of the their related integral representations are obtained in terms of higher transcendental hypergeometric functions like Lauricella’s hypergeometric function FC(m)[x], Fox–Wright Ψ function, Srivastava–Daoust S generalized Lauricella function, Riemann Zeta and Dirichlet Eta functions, while the extensions concern products of Bessel and modified Bessel functions of the first kind, hyper–Bessel and Bessel–Clifford functions. Auxiliary Laplace–Mellin transforms, bounding inequalities for the hyper–Bessel and Bessel–Clifford functions are established- which are also of independent but considerable interest. A set of bounding inequalities are presented either for the hyper–Bessel and Bessel–Clifford functions which are to our best knowledge new, or also for all considered extended Mathieu–type series. Next, functional bounding inequalities, log–convexity properties and Turán inequality results are presented for the investigated extensions of multi–parameter Mathieu–type series. We end the exposition by a thorough discussion closes the exposition including important details, bridges to occuring new questions like the similar kind multi–parameter treatment of the complete Butzer–Flocke–Hauss Ω function which is intimately connected with the Mathieu series family.