Extension of Mathieu series and alternating Mathieu series involving the Neumann function Y_nu

Abstract

The main objective of this paper is to present a new extension of the familiar Mathieu series and the alternating Mathieu series S(r) and {{widetilde{S}}}(r) which are denoted by {mathbb {S}}_{mu ,nu }(r) and widetilde{{mathbb {S}}}_{mu ,nu }(r), respectively. The computable series expansions of their related integral representations are obtained in terms of the exponential integral E_1, and convergence rate discussion is provided for the associated series expansions. Further, for the series {mathbb {S}}_{mu ,nu }(r) and widetilde{{mathbb {S}}}_{mu ,nu }(r), related expansions are presented in terms of the Riemann Zeta function and the Dirichlet Eta function, also their series built in Gauss’ {}_2F_1 functions and the associated Legendre function of the second kind Q_mu ^nu are given. Our discussion also includes the extended versions of the complete Butzer–Flocke–Hauss Omega functions. Finally, functional bounding inequalities are derived for the investigated extensions of Mathieu-type series.