Abstract
This study is based on a part of the results obtained in the author’s publications. An enumerable family of the Le Roy type functions is considered herein. The asymptotic formula for these special functions in the cases of ‘large’ values of indices, that has been previously obtained, is provided. Further, series defined by means of the Le Roy type functions are considered. These series are studied in the complex plane. Their domains of convergence are given and their behaviour is investigated ‘near’ the boundaries of the domains of convergence. The discussed asymptotic formula is used in the proofs of the convergence theorems for the considered series. A theorem of the Cauchy–Hadamard type is provided. Results of Abel, Tauber and Littlewood type, which are analogues to the corresponding theorems for the classical power series, are also proved. At last, various interesting particular cases of the discussed special functions are considered.
Highlights
In the series of papers [7,8,9,10], as well as in the recent book [11], we studied series in systems of some representatives of the special functions of fractional calculus, which are fractional index analogues of the Bessel functions and multi-index Mittag–Leffler functions, and we have proved various results connected with their convergence in the complex domains
Let us note that the series (11) absolutely converges in the open disk D (0; R) with the radius R, given by (12), and it diverges in its outside, like in the classical theory of the power series
N =0 in this case coinciding with the series (11) in Le Roy type functions with complex coefficients an (n = 0, 1, 2, . . . ) and for z ∈ C
Summary
As is established in [3], this function turns out to be an entire function of the complex variable z for all values of the parameters such that. Hadamard fractional derivatives or hyper-Bessel-type operators see Garra-Polito [2], different integral representations can be seen in [3] and Pogány [6]. In this paper, considering the Le. Roy type functions (1), we discuss various earlier results which are needed here. Roy type functions (1), we discuss various earlier results which are needed here These are results related to inequalities in the complex plane C and on its compact subsets and asymptotic formula for ‘large’ values of indices of the functions (1). In the series of papers [7,8,9,10], as well as in the recent book [11], we studied series in systems of some representatives of the special functions of fractional calculus, which are fractional index analogues of the Bessel functions and multi-index Mittag–Leffler functions (in the sense of [12,13,14,15]), and we have proved various results connected with their convergence in the complex domains
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