Extension of Mathieu series and alternating Mathieu series involving the Neumann function $$Y_\nu $$

Abstract

AbstractThe main objective of this paper is to present a new extension of the familiar Mathieu series and the alternating Mathieu series S(r) and $${{\widetilde{S}}}(r)$$ S ~ ( r ) which are denoted by $${\mathbb {S}}_{\mu ,\nu }(r)$$ S μ , ν ( r ) and $$\widetilde{{\mathbb {S}}}_{\mu ,\nu }(r)$$ S ~ μ , ν ( r ) , respectively. The computable series expansions of their related integral representations are obtained in terms of the exponential integral $$E_1$$ E 1 , and convergence rate discussion is provided for the associated series expansions. Further, for the series $${\mathbb {S}}_{\mu ,\nu }(r)$$ S μ , ν ( r ) and $$\widetilde{{\mathbb {S}}}_{\mu ,\nu }(r)$$ S ~ μ , ν ( r ) , related expansions are presented in terms of the Riemann Zeta function and the Dirichlet Eta function, also their series built in Gauss’ $${}_2F_1$$ 2 F 1 functions and the associated Legendre function of the second kind $$Q_\mu ^\nu $$ Q μ ν 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.