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ABSTRACT. In this paper we consider a family of 3-index generalizations of the classical Mittag-Leffler functions. We study the convergence of series in such functions in the complex plane. First we find the domains of convergence of such series and then study their behaviour on the boundaries of these domains. More precisely, Cauchy-Hadamard, Abel, Tauber and Littlewood type theorems are proved as analogues of the classical theorems for the power series.

Abstract:In this paper we consider a family of 3 -indices generalizations of the classical Mittag-Leffler function, called multi-index (3 -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 Ψ -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. PACS (2008):

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