Le Roy Functions and Series in Them Inequalities Asymptotic Formula Convergence of Power and Functional Series in Complex Plane Cauchy–Hadamard, Abel, Tauber and Littlewood Type Theorems
Issue Date:
12-Jun-2021
Publisher:
MDPI
Citation:
Paneva-Konovska, J. Series in Le Roy Type Functions: A Set of Results in the Complex Plane-A Survey. Mathematics, 2021, 9, 1361. https://doi.org/10.3390/math9121361
Series/Report no.:
Mathematics;9, 1361
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.
