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Integral representations and integral transforms of some families of Mathieu type series

By using some integral representations for several Mathieu type series (see [2], [3], [10], [15], and [17]) via the Bessel function $J_\nu$ of the first kind, the Gauss hypergeometric function $_2F_1$, the generalized hypergeometric function $_pF_q$ and the Fox-Wright generalization $_p\Psi_q$ of the hypergeometric function $_pF_q$, a number of integral representations of the Laplace, Fourier, and Mellin types are derived here for certain general families of Mathieu type series. Some interesting corollaries and consequences of these integral representations are also considered.

Mathieu type series; integral representation; Laplace; Fourier; Mellin transform; generalized hypergeometric functions; Fox-Wright Psi function

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