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Integral expression of Mathieu - type series whose terms contain Fox's $H$ - function

Closed integral form expressions are derived for the Mathieu--type $\mathbf a$--series and for the associated alternating series whose terms contain Fox's $H$--function. In the final section, two similar results on Mathieu''tzpe series with Fox--Wright Psi--function $\, _p\Psi_q$ and the generalized hypergeometric function $\, _pF_q$ which can be expressed in terms of the Fox's $H$--functions are deduced respectively.

Dirichlet series; Fox's $H$ function; Fox--Wright $\Psi$ function; hypergeometric $\; _pF_q$ function; integral representation of series; Laplace integral representation of Dirichlet series; Mathieu--type $\mathbf a$--series

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