Two special functions, generalizing the Mittag–Leffler type function, in solutions of integral and differential equations with Riemann-Liouville and Kober operators

Two special functions, concerning Mittag–Leffler type functions, are considered. The first is the modification of generalized Mittag–Leffler type function, introduced by A. A. Kilbas and M. Saigo; the second is the special case of the first one. The solutions of the integral equation with the Kober operator and the generalized power series as the free term are presented. The existence and uniqueness of these solutions are proved. The explicit solutions of the integral equations above are found out in terms of introduced special functions. The correctness of initial value problems for linear homogeneous differential equations with Riemann–Liouville and Kober fractional derivatives is investigated. The solutions of the Cauchy type problems are found out in the special classes of functions with summable fractional derivative via the reduction to the considered above integral equation and also are written in the explicit form in terms of the introduced special functions. The replacement of the Cauchy type initial values to the modified (weight) Cauchy conditions is substantiated. The particular cases of parameters in the differential equations when the Cauchy type problems are not well-posed in sense of the uniqueness of solutions are considered. In these cases the unique solutions of the Cauchy weight problems are existed. It is noted in this paper that the weight Cauchy problems allow to expand the acceptable region of the parameters values in the differential equations to the case when the fractional derivative has the nonsummable singularity in zero.