engleski

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

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, respectively, by \mathbb S_{; ; \mu, \nu}; ; (r) and \widetilde{; ; \mathbb S}; ; _{; ; \mu, \nu}; ; (r). The computable series expansions of their related integral representations are obtained in terms of 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 Dirichlet Eta function, also their series built in Gauss' 2F_1 functions and associated Legendre function of the second kind Q_mu^nu are given. 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.

Mathieu and alternating Mathieu series ; Neumann function Y_nu ; Euler-Abel transformation of series ; Exponential integral E_1 ; Gubler-Weber formula ; Associated Legendre function of second kind ; Riemann Zeta function ; Dirichlet Eta function ; Polylogarithm ; Complete Butzer-Flocke-Hauss Omega function ; Functional bounding inequality

nije evidentirano