Integral form of Mathieu-type (a, A; α )–series which contains fractional powers special functions

We aim to give closed integral form expressions for the Mathieu-type (a, A; α ) ( a , A ; α ) -series and their alternating counterparts, which contain fractional powers special functions as generalized Fox-Wright hypergeometric function aₚΨ q a p Ψ ~ q in their terms, together with their contiguous representations with respect to the additional parameters. Particular cases of aₚΨ q a p Ψ q are specified, including the Le Roy–type functions and their multi–index analogues and other special functions. The corresponding results are illustrated with important special cases.