On two special functions, generalizing the Mittag-Leffler type function, their properties and applications

Two special functions, concerning Mittag-Leffler type functions, are studied. The first is the modification of generalized Mittag–Leffler function, which was introduced by A. A. Kilbas and M. Saigo; the second is the special case of the first one. The relation of these functions with some elementary and special functions and their role in solving of Abel–Volterra integral equations is indicated. The formulas of the fractional integration and differentiation in sense of Riemann–Liouville and Kober are presented. The applications to Cauchy type problems for some linear fractional differential equations with Riemann-Liouville and Kober derivatives are noticed.