Abstract
The purpose of this paper is to develop some new recurrence relations for the two parametric Mittag-Leffler function. Then, we consider some applications of those recurrence relations. Firstly, we express many of the two parametric Mittag-Leffler functions in terms of elementary functions by combining suitable pairings of certain specific instances of those recurrence relations. Secondly, by applying Riemann–Liouville fractional integral and differential operators to one of those recurrence relations, we establish four new relations among the Fox–Wright functions, certain particular cases of which exhibit four relations among the generalized hypergeometric functions. Finally, we raise several relevant issues for further research.
Highlights
Introduction andPreliminaries10.3390/fractalfract5040215Magnus Gösta Mittag-Leffler (1846–1927), a Swedish mathematician, invented the function Eμ (z) (1) in conjunction with the summation technique for divergent series, which is eponymously referred to as the Mittag-Leffler (M-L) function and represented by the following convergent power series across the whole complex plane: Academic Editor: Maja AndrićReceived: 19 October 2021 ∞Accepted: 10 November 2021 Eμ (z) =Published: 12 November 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Licensee MDPI, Basel, Switzerland.E1 (±z) = e±z, distributed under the terms and conditions of the Creative Commons creativecommons.org/licenses/by/
We recall some of the recurrence relations for the two parametric Mittag-Leffler function (5) and the three parametric Mittag-Leffler function (7)
Salim ([60], [Theorem 2.2]) presented two interesting differential recurrence relation for an extended Mittag-Leffler function, which can be derived by replacing n! in (7) with a Pochhammer symbol, say (η )n under the constraint 0
Summary
Magnus Gösta Mittag-Leffler (1846–1927), a Swedish mathematician, invented the function Eμ (z) (1) in conjunction with the summation technique for divergent series, which is eponymously referred to as the Mittag-Leffler (M-L) function and represented by the following convergent power series across the whole complex plane (see [1,2,3,4]): Academic Editor: Maja Andrić. Numerous mathematicians have investigated the properties of this entire function (see, e.g., [7,8,9,10,11,12,13,14,15,16]) This function (1) reduces to a number of elementary and special functions such as (see, e.g., ([6], [Section 3.2])). The Mittag-Leffler function (1), its slight generalization (5), the Prabhakar function (7), and a number of other parameterized extensions are found to be particular instances of the following Fox–Wright function defined by We recall some of the recurrence relations for the two parametric Mittag-Leffler function (5) and the three parametric Mittag-Leffler function (7)
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