Generalized Mittag-Leffler Type Function: Fractional Integrations and Application to Fractional Kinetic Equations

The generalized fractional integrations of the generalized Mittag-Leffler type function (GMLTF) are established in this paper. The results derived in this paper generalize many results available in the literature and are capable of generating several applications in the theory of special functions. The solutions of a generalized fractional kinetic equation using the Sumudu transform is also derived and studied as an application of the GMLTF.


INTRODUCTION
The Pochhammer symbol (̟ ) n is defined by (for ̟ ∈ C)[see ( [1], p. 2 and p. The familiar generalized hypergeometric function p F q is defined as follows (see [2]): (p ≤ q, x ∈ C; p = q + 1, |x| < 1), where (̟ j ) n and (χ j ) n given in (1) and χ i can not be a negative integer or zero. Here p or q or both are permitted to be zero. For all finite x, the series (2) is absolutely convergent if p ≤ q and for |x| < 1 if p = q + 1. When p > q + 1, then the series diverge for x = 0 and the series does not terminate.

GENERALIZED FRACTIONAL INTEGRATION OF GMLTF
Fractional calculus is one of the prominent branch of applied mathematics that deals with non-integer order derivatives and integrals (including complex orders), and their applications in almost all disciplines of science and engineering [18][19][20][21][22]. In this line, the use of special functions in connection with fractional calculus also studied widely [23][24][25][26][27]. For the basics of fractional calculus and its related literature, interesting readers can be referred to as Kiryakova [28], Miller and Ross [29], and Srivastava et al. [30]. In this paper, we studied the generalized fractional calculus of more generalized function given in (7). The generalized fractional integral operators (FIOs) involving the Appell functions F 3 are given for ̟ , ̟ ′ , τ , τ ′ , ǫ ∈ C with ℜ(ǫ) > 0 and x ∈ R + as follows: and The integral operators of the types (8) and (9) have been introduced by Marichev [31] and later extended and studied by Saigo and Maeda [32]. Recently, many researchers (see [33][34][35]) have studied the image formulas for MSM FIOs involving various special functions. The corresponding fractional differential operators (FDOs) have their respective forms: f (x) (10) and Here, we recall the following results (see [32,36]).
Then there exists the relation Frontiers in Physics | www.frontiersin.org The main aim of this paper is to apply the generalized operators of fractional calculus for the GMLTF in order to get certain new image formulas.

Sumudu Transform
The Sumudu transform is widely used to solve various type of problems in science and engineering and it is introduced by Watugala (see [37,38]). The details of Sumudu transforms, properties, and its applications the interesting readers can be refer to Asiru [39], Belgacem et al. [40], and Bulut et al. [41]. The Sumudu transform over the set functions is defined by The main aim of this study is to establish the generalized fractional calculus operators and the generalized FKEs involving GMLTF.
The following corollaries can derive immediately from Theorems 2.1 and 2.2 with the help of Pochhammer symbol In the next section, we derived the generalized FKEs and we consider the Sumudu transform methodology to achieve the results.

GENERALIZED FRACTIONAL KINETIC EQUATIONS INVOLVING GMLTF
The generalized fractional kinetic equations (FKEs) involving the GMLTF with the Sumudu transform is derived in this section. The FKEs are studied widely in many papers [42][43][44][45]. Let K = (K t ) be the arbitrary reaction defined by a timedependent quantity. The destruction d and production p depend on the quantity K itself: d = d(K) or p = p(K) [see [42]]. The fractional differential equation can be expressed by [42]). A special case of (15) is with K i (t = 0) = K 0 , c i > 0 and the solution of (16) is Performing the integration of (16) leads to where 0 D −1 t is the particular case of Riemann-Liouville (R-L) integral operator and c is a constant. The fractional form of (18) is (see [42]) where 0 D −µ t is given by Theorem 3.1. For ̟ , λ, γ ∈ C, δ = 0, −1, −2, · · · , d > 0, ǫ > 0 then the solution of is given by Proof: The Sumudu transform (ST) of the R-L fractional operator is where G (u) is defined in (14). Now, applying the ST on the both sides of (21) and using (7) and (23), we have which gives which implies that After some simple calculation, we get The inverse ST of (27) and using the formula In view of the Mittag-Leffler function definition, we arrived the needful result.

CONCLUSION
The generalized fractional integrations of the generalized Mittag-Leffler type function is studied in this paper. The obtained results are expressed in terms of the generalized Wright hypergeometric function and generalized hypergeometric functions. To show the potential application of GMLTF, the solutions of fractional kinetic equations are derived with the help of Sumudu transform.
The results obtained in this study have significant importance as the solution of the equations are general and can derive many new and known solutions of FKEs involving various type of special functions.

DATA AVAILABILITY STATEMENT
All datasets generated for this study are included in the article.

AUTHOR CONTRIBUTIONS
The author confirms being the sole contributor of this work and has approved it for publication.