On the multi-index (3m-parametric) Mittag-Leffler functions, fractional calculus relations and series convergence

In this paper we consider a family of 3m-indices generalizations of the classical Mittag-Leffler function, called multi-index (3m-parametric) Mittag-Leffler functions. We survey the basic properties of these entire functions, find their order and type, and new representations by means of Mellin-Barnes type contour integrals, Wright pΨq-functions and Fox H-functions, asymptotic estimates. Formulas for integer and fractional order integration and differentiations are found, and these are extended also for the operators of the generalized fractional calculus (multiple Erdélyi-Kober operators). Some interesting particular cases of the multi-index Mittag-Leffler functions are discussed. The convergence of series of such type functions in the complex plane is considered, and analogues of the Cauchy-Hadamard, Abel, Tauber and Littlewood theorems are provided.