Naslov
Some families of Mathieu $\mathbf a$-series and alternating Mathieu $\mathbf a$-series

Autori
Poganj, Tibor K. ; Srivastava, Hari M. ; Tomovski, Živorad

Izvornik
Applied mathematics and computation (0096-3003) 173 (2005), 1; 69-108

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Mathieu and alternating Mathieu series; Mathieu and alternating Mathieu $\mathbf a$-series; Mellin transform; Asymptotic expansions; Bernoulli numbers; Bessel function of the first kind; Dirichlet Eta function; Dirichlet series; Euler-Maclaurin etc.

Sažetak
The main purpose of ths paper is to present a number of potentially useful intergal representations for the familiar Mathieu $\mathbf a$-series as well as for its alternating version. These results are derived here from many different considerations and are shown to yield sharp bounding inequalities involving the Mathieu and alternating Mathieu $\mathbf a$-series. Relationships of the Mathieu $\mathbf a$-series with the Reimann Zeta and the Dirichlet Eta function are also considered. Such special functions as the classical Bessel function $J_\nu(z)$ and the confulent hypergeometric functions $_0F_1$ and $_1F_2$ are characterized by means of certain Fredholm type intergal equations of the first kind, which are associated with soem of these Mathieu type series. Several intergals containing Mathieu type series are aslo evaluated. Finally, some closely-realted new questions and open problems are indicated with a view to motivating further investigations on the subject of this paper.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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