engleski

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

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.

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.

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