Some new integral representations of generalized Mathieu series and alternating Mathieu series

Abstract

The main purpose of this paper is to present a number of new integral representations for the familiar Mathieu series $S_\mu^{(\alpha,\beta )}(r;\{a_k\}_{k=1}^\infty)(r\in R$, $\alpha$, $\beta$, $\mu$, $\{a_k\}_{k=1}^\infty\in R^+)$ [12]  as well as for its alternating version [8,16] when $a_k=\{k^p\}_{k=1}^\infty$, $a_k=\{(k!)^p\}_{k=1}^\infty$, $a_k=\{(\ln k!)^p\}_{k=1}^\infty$ with $p=\gamma$, $\gamma(\mu\alpha-\beta)>1$ and $p=\frac q\alpha$, $\mu -\frac \beta\alpha>q^{-1}$, $q\in N$.