On some properties of the generalized Mittag-Leffler function

Communicated by: W. Sprößig MSC Classification: Primary 26A33; 33C60; 33E12; Secondary 33E20; 44A40; 45J05Motivated by the demonstrated potential for their applications in various research areas such as those in mathematical, physical, engineering, and statistical sciences, our main object in this paper is to introduce and investigate a fractional integral operator that contains a certain generalized multi-index Mittag-Leffler function in its kernel. In particular, we establish some interesting expressions for the composition of such well-known fractional integral and fractional derivative operators as (for example) the Riemann-Liouville fractional integral and fractional derivative operators, the Hilfer fractional derivative operator, and the above-mentioned fractional integral operator with the generalized multi-index Mittag-Leffler function in its kernel. The main findings in this paper are shown to generalize the results that were derived earlier by Kilbas et al 27 and Srivastava et al. 9 Finally, in this paper, we derive integral representations for the product of 2 generalized multi-index Mittag-Leffler functions in terms of the familiar Fox-Wright hypergeometric function. KEYWORDS fractional derivative operators, Fox-Wright hypergeometric function, generalized multi-index Mittag-Leffler function, integral representations, Lebesgue measurable functions, Rice, Jacobi, and related hypergeometric polynomials in 1 and 2 variables

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“…We also recall the following 2 generalizations of the Mittag‐Leffler–type functions:

E_{α, β, p}^{η, δ, q} false(z false) = true \sum_{n = 0}^{\infty} \frac{{false(η false)}_{q n}}{normalΓ false(α n + β false)} \frac{z^{n}}{{false(δ false)}_{p n}}

(p, q \in {double-struckR}^{+}; α, β, η, δ \in C; ℜ (α) > 0)

and

E_{α, β, ν, σ, δ, p}^{μ, ρ, η, q} false(z false) = true \sum_{n = 0}^{\infty} \frac{{false(μ false)}_{ρ n} {false(η false)}_{q n}}{{false(ν false)}_{σ n} {false(δ false)}_{p n}} \frac{z^{n}}{normalΓ false(α n + β false)}

(p, q \in {double-struckR}^{+}; q ≦ ℜ (α) + p; α, β, η, δ, μ, ν, ρ, σ \in C; \min {ℜ (α), ℜ (ρ), ℜ (σ)} > 0) .

…”

Section: Introduction Definitions and Preliminariesmentioning

confidence: 99%

A study of fractional integral operators involving a certain generalized multi‐index Mittag‐Leffler function

Srivástava

Bansal

Harjule

2018

Math Methods in App Sciences

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“…The importance and great considerations of Mittag-Leffler functions have led many researchers in the theory of special functions to exploring possible generalizations and applications. Many more extensions or unifications for these functions are found in a large number of papers [1][2][3][4][5]. A useful generalization of the Mittag-Leffler function, the so-called Mittag-Leffler k-function has been introduced and studied in [6].…”

Section: Introductionmentioning

confidence: 99%

Fractional operators with generalized Mittag-Leffler k-function

Mubeen

Ali

2019

Adv Differ Equ

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In this paper, our main aim is to deal with two integral transforms involving the Gauss hypergeometric functions as their kernels. We prove some composition formulas for such generalized fractional integrals with Mittag-Leffler k-function. The results are established in terms of the generalized Wright hypergeometric function. The Euler integral k-transformation for Mittag-Leffler k-functions has also been developed.

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“…The importance and great considerations of MittagLeffler function have led many researchers in the theory of special functions for exploring the possible generalizations and applications. Many more extensions or unifications for these functions are found in large number of papers [6,[21][22][23][24]. A useful generalization of the Mittag-Leffler function called as k-MittagLeffler function E γ k,α,β (z), introduced in [4], and it is given by 4) where α, β, γ ∈ C, k ∈ R, { (α) , (β) , (γ)} > 0 and (γ) n,k is the k-Pochhammer symbol defined as:…”

Section: Introductionmentioning

confidence: 99%

“…Lately, a generalized form of k-Mittag-Leffler function was introduced and studied in [5] as: 6) where (γ) nq,k is defined as (1.5) and the generalized Pochhammer symbol is defined as (see [19]):…”

Section: Introductionmentioning

confidence: 99%