Closed form expressions are obtained for a family of convergent Mathieu type a-series and its alternating variants, whose terms contain an I-function which is a generalization of the Fox's H-function. The results derived are of general character and provide an elegant generalization for the closed form expressions of these series associated with the H-function by Pogány, for Fox-Wright functions by Pogány and Srivastava and for pFq and Meijer's G-function by Pogány and Tomovski, and others.