Fractional Calculus of Extended Mittag-Leffler Function and Its Applications to Statistical Distribution

Several fractional calculus operators have been introduced and investigated. In this sequence, we aim to establish the Marichev-Saigo-Maeda (MSM) fractional calculus operators and Caputo-type MSM fractional differential operators of extended Mittag-Leffler function (EMLF). We also investigate the statistical distribution associated with the EMLF. Finally, we derive some of the particular cases of the main results.


Introduction and Preliminaries
Fractional calculus (FC) is a discipline of mathematics that derives from the conventional definitions of integral and derivative operators by considering fractional values. The reason for attracting the scientist towards FC is that fractional derivatives have been recognized as powerful modeling and simulation tools for engineering problems. Many physical laws are expressed more accurately in terms of differential equations of arbitrary order. The fractal calculus can efficiently deal with kinetics, which is termed the fractal kinetics [1][2][3]. The Mittag-Leffler (M-L) function and its generalizations are widely used in the field of fractals. The generalized M-L law with fractal calculus appears in [4]. The use of M-L function in the medical field with fractals is given in [5]. In [6], authors defined the M-L function on fractal sets. For more details about the use of the Mittag-Leffler function in the field of fractal calculus and applications, interested readers can refer to [7][8][9]. FC has potential applications in the variational iteration method (VIM). In [10], authors used the local fractional operators to investigate the application of local fractional VIM for solving the local fractional Laplace equations. A new VIM for a class of fractional convection-diffusion equations is given in [11]. Numerous papers on VIM and its various applications are found in many research articles [12][13][14].
Sharma and Devi [26] introduced and investigated the following extended Wright generalized hypergeometric function where the empty product is understood to be 1 and when the summation is assumed to be convergent. We recall the fractional integral operators with the Appell function F 3 (see, e.g., [27], p. 53, Equation (6)) as a kernel (see [28,29]): The generalized fractional integral operators involving the Appell functions F 3 are defined for ν, ν , ξ, ξ , ϑ ∈ C with (ϑ) > 0 and x ∈ R + as follows: and The integral operators of the types (9) and (10) have been introduced by Marichev [28] and later extended and studied by Saigo and Maeda [29]. Recently, many researchers (see [30][31][32][33][34][35][36]) have studied the image formulas for MSM fractional integral operators involving various special functions.
The corresponding fractional differential operators have their respective forms: f (x) (11) and Here, we recall the following lemmas (see [29,37]).
then there exists the relation The left-and right-sided generalized integral transforms defined for x > 0 and ν, ξ, ϑ ∈ C, (ν) > 0, respectively by (see [38]) and The left-and right-hand-sided Riemann-Liouville fractional integrals of order ν ∈ C are defined by and Also, we need the following lemmas [38].
In particular, and In particular, and The generalized forms of the M-L function and its properties have appeared in recent papers [39][40][41]. The objective of this paper is to present generalized fractional integral and differential operators of EMLF and their application to statistical distribution. The presented work is arranged as follows: In Sections 2 and 3, a form of MSM fractional integral and differential representations of (5) is presented alongside its properties. In Section 4, Caputo-type MSM fractional differential operators are discussed.
In Section 5, we also presented some statistical distribution regarding (5) and conclusion drawn in Section 6.

MSM Fractional Integral Representations of Extended Mittag-Leffler Function
Here we present generalized EMLF in view of the MSM fractional integral representations and consider some particular cases.
Proof. Let S 1 be LHS of (24), then using (5), we have Interchanging summation and integration order which is verified under the condition in this theorem, we get Applying the Lemma 1, we get Thus, by using (8), we get the result.

MSM Fractional Differential Representations of Extended Mittag-Leffler Function
In this part, we present the MSM fractional differentiation of (5). We recall the following lemmas (see [37]).
Proof. Let S 4 be LHS of (31), then using (5), we have Interchanging summation and integration order i.e., verified under the condition in this theorem, we get Applying the Lemma 6, we get x n n! Again, by using (8), we get the desired result.

Caputo-Type MSM Fractional Differentiation of Extended Mittag-Leffler Function
In this part, we discuss the left-and right-hand-sided Caputo-type fractional derivatives that have the Gauss hypergeometric function in the kernel are given as: and where ν, ξ, ϑ, ∈ C and x ∈ R + . The left-and right-hand-sided Caputo-type MSM fractional differential operators: (34) and where ν, ν , ξ, ξ , ϑ, ∈ C and x ∈ R + . To discuss the Caputo-type MSM fractional differential operator of the extended MLF (5), the following lemmas will be required to prove the proposed result.
Proof. let S 6 be L. H. S. of (37), then using (5), we have Interchanging summation and integration order, which is verified under the given condition, we get Applying the Lemma 8, we get x n n! .
By using (8), we get the required result.

Extended Mittag-Leffler Function and Statistical Distribution
For a random variable X, the distribution function is defined by where x is any real number −∞ < x < ∞. The properties of distribution function F(x) as follows is continuous from the right. Many authors studied the distribution function which involves the M-L function [41][42][43][44]. In this line, we develop the distribution function involving extended M-L function (5) and deduce particular cases of our result.

Concluding Remarks
FC operators have significant applications in the field of science and engineering. Many research papers have been used to solve nonlinear differential equations, VIMs, and fractal related problems with the help of fractional operators. In this lineage, we established generalized fractional formulas to derive numerous results. The operators developed in this paper may have applications in applied mathematics and physics. The significant generality of these results rendered some existing results as particular cases of our result. For instance, if we let p = 0, then we obtain MSM fractional integral, MSM fractional differential formulas and Caputo-type MSM fractional differentiation formulas of Mittag-Leffler function defined in (3) (see [34]).
Author Contributions: All authors contributed equally to this manuscript.
Funding: This research received no external funding.