Convergence of series in three parametric Mittag-Leffler functions

Abstract

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.

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

“…Here we recall some results in this direction related to the Prabhakar (3-parameters Mittag-Leffler type) functions (2) and for the (2 ) multi-index M-L functions (5), including asymptotic formulae for "large" values of indices of these functions, obtained in [27,30,31].…”

“…Furthermore, an asymptotic formula for "large" values of the indices is valid as follows, for a proof see [31]. …”

Generalized Langevin Equation and the Prabhakar Derivative

“…Such series in three parameter M-L functions are convergent [36][37][38]. From Equation (22), for the long time limit we find…”

Subordination Principle for a Class of Fractional Order Differential Equations

“…Other kinds of asymptotic estimates were provided in [29], and on their basis, the convergence of series in Functions (10) in the complex domain, similar to these appearing further in (27) and (34), is studied in the recent papers [25,30].…”