Mathieu series and associated sums involving the Zeta functions

Almost twelve decades ago, Mathieu investigated an interesting series S(r)S(r) in the study of elasticity of solid bodies. Since then many authors have studied various problems arising from the Mathieu series S(r)S(r) in various diverse ways. In this paper, we present a relationship between the Mathieu series S(r)S(r) and certain series involving the Zeta functions. By means of this relationship, we then express the Mathieu series S(r)S(r) in terms of the Trigamma function ψ′(z)ψ′(z) or (equivalently) the Hurwitz (or generalized) Zeta function ζ(s,a)ζ(s,a). Accordingly, various interesting properties of S(r)S(r) can be obtained from those of ψ′(z)ψ′(z) and ζ(s,a)ζ(s,a). Among other results, certain integral representations of S(r)S(r) are deduced here by using the aforementioned relationships among S(r)S(r), ψ′(z)ψ′(z) and ζ(s,a)ζ(s,a).