Generalized Mittag-Leffler function and generalized fractional calculus operators

The paper is devoted to the study of the function E-rho,mu(gamma) (z) defined for complex rho, mu, gamma (Re(rho) > 0) by [GRAPHICS] which is a generalization of the classical Mittag-Leffler function E-rho,mu(gamma)(z) and the Kummer confluent hypergeometric function Phi(gamma, mu; z ). The properties of E-rho,mu(gamma)(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel (E(rho,mu,omega;a+)(gamma)phi)(x) =integral(a)(x) (x-t)E-mu-1(rho,mu)gamma[omega(x-t)(rho)]phi(t)dt (x>a) is studied in the space L (a,b). Compositions of the Riemann-Liouville fractional integration and differentiation operators with E-rho,mu,omega;a+(gamma) are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E-rho,mu,omega;a+(gamma). Since, for gamma = 0, E-rho,mu,omega;a+(0) coincides with the Riemann-Liouville fractional integral of order mu, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mittag-Leffler function E-rho,mu(gamma)(z) in the kernels. Similar assertions are presented for the integral operators containing the Mittag-Leffler and Kummer functions, E-rho,mu(gamma)(z) and Phi(gamma, mu; z ), in the kernels, and applications are given to obtain solutions in closed form of the integral equations of the first kind.