Multivariate analogue of generalized Mittag-Leffler function

Abstract

Following the results of Saxena and Kalla [Solutions of Volterra-type integro-differential equations with a generalized Lauricella confluent hypergeometric function in the kernels, Int. J. Math. Math. Sci. 8 (2005), pp. 1155–1170], we introduce and develop here a theory of multivariate generalization of the Mittag-Leffler function, which is defined as: where λ, γ j , ρ j ∈C, Re (ρ j )>0, j=1, …, m. Certain properties of this multivariate generalized Mittag-Leffler function associated with fractional calculus are established. Further, an integral operator with this function as a kernel, in the following form: is studied in the space L(a, b). An analogy of the semi-group property for the composition of two such operators with different indices is proved. A composition of the Riemann–Liouville fractional integral operator with two such operators with different indices is established. The results derived in this paper provide generalization of the results given earlier by Kilbas et al. [Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms Spec. Funct. 15 (2004), 31–49].