Mathieu series and associated sums involving the Zeta functions

Almost twelve decades ago, Mathieu investigated an interesting series 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/ in various diverse ways. In this paper, we present a relationship between the Mathieu series S.r/ and certain series involving the Zeta functions. By means of this relationship, we then express the Mathieu series S.r/ in terms of the Trigamma function 0.z/ or (equivalently) the Hurwitz (or generalized) Zeta function .s; a/. Accordingly, various interesting properties of S.r/ can be obtained from those of 0.z/ and .s; a/. Among other results, certain integral representations of S.r/ are deduced here by using the aforementioned relationships among S.r/, 0.z/ and .s; a/.