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Integral expressions for Mathieu-type series of Fox's $H$-Function Terms and Related Consequences

In some recent articles Pog\'any, Srivastava and Tomovski consider special kind Mathieu-type series and their alternating variants whose terms contain the familiar Gau{;\ss}; hypergeometric function $\, _2F_1$, generalized hypergeometric function $\, _pF_q$ and Meijer's $G$ function. The derived results concern among others closed integral form expressions for the considered series and bilateral bounding inequalities. At this address we are interested in generalizing the expression results for Mathieu-type series (and its alternating variants) whose terms contain Fox's $H$-function. Then using the well-known reduction formul{;\ae}; existing between the $H$-, and $G, \, _pF_q$-functions we deduce closed form integral epxression for the considered Mathieu-type series.

Mathieu-type series; Generalized hypergeometric functions; Fox-Wright $\Psi$ function; Meijer's $G$-function; Fox's $H$-function; Integral representation

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