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 the familiar Mathieu series and alternating S ( r ) $${{\widetilde{S}}}(r)$$ S ~ ( r ) which are denoted by $${\mathbb {S}}_{\mu ,\nu }(r)$$ μ , ν $$\widetilde{{\mathbb {S}}}_{\mu , respectively. computable expansions their related integral representations obtained in terms exponential $$E_1$$ E 1 convergence rate discussion provided for associated expansions. Further, presented Riemann Zeta function Dirichlet Eta function, also built Gauss’ $${}_2F_1$$ 2 F functions Legendre second kind $$Q_\mu ^\nu $$ Q given. Our includes extended versions complete Butzer–Flocke–Hauss Omega functions. Finally, functional bounding inequalities derived investigated extensions Mathieu-type series.