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 $${{\widetilde{S}}}(r)$$ 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.

Applications of Euler Sums and Series Involving the Zeta Functions

“…Since Mathieu's era, numerous researchers have explored various facets of the Mathieu series, such as (94), as well as (95), in a range of techniques (see, for instance, [55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74]).…”