Some families of Mathieu a-series and alternating Mathieu a-series

The main purpose of this paper is to present a number of potentially useful integral representations for the familiar Mathieu 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 a-series. Relationships of the Mathieu a-series with the Riemann Zeta function and the Dirichlet Eta function are also considered. Such special functions as the classical Bessel function Jν(z) and the confluent hypergeometric functions 0F1 and 1F2 are characterized by means of certain Fredholm type integral equations of the first kind, which are associated with some of these Mathieu type series. Several integrals containing Mathieu type series are also evaluated. Finally, some closely-related new questions and open problems are indicated with a view to motivating further investigations on the subject of this paper.