The New Mittag-Leffler Function and Its Applications

Department of Mathematics, University of Sargodha, Sargodha, Pakistan Department of Mathematics, Prince Sultan University, Riyadh, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung 40402, Taiwan Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawaser, Saudi Arabia


Introduction
e theory of special functions comprises a major part of mathematics. In the last three centuries, the essential of solving the problems taking place in the fields of classical mechanics, hydrodynamics, and control theories motivated the development of the theory of special functions. is field also has wide applications in both pure mathematics and applied mathematics. e interested readers may consult the literature [1][2][3][4].
e Mittag-Leffler function takes place naturally similar to that of the exponential function in the solutions of fractional integro-differential equations having the arbitrary order. e Mittag-Leffler functions have to gain more recognition due to its wide applications in diverse fields. We suggest the readers to review the literature [5][6][7][8][9][10][11][12][13][14][15][16] for more details.
roughout this article, let C, R + , Z − , and N be the sets of complex numbers, positive real numbers, negative integers, and natural numbers, respectively.

Preliminaries
is section contains some basic definitions and mathematical preliminaries. We begin with the well-known Mittag-Leffler function.
Definition 5. For p, k, ξ ∈ R, 0 < s < 1, and n ∈ N, where e following identities are satisfied: Definition 6. In [23], the gamma (p, s, k)-function in term of s-series is given by e relationship between p [ξ] n,k,s and p Γ s,k (ξ) is given by [23] e beta (p, k)-function is defined by Definition 7. e beta (p, k)-function is represented by the following integral: Definition 8. e well-known Laplace transform of piecewise continuous function f: R ⟶ R is defined by

Applications and Properties of the Pochhammer (p, s, k)-Symbol p [ξ] n,k,s and Gamma (p, s, k)-Function p Γ s,k (ξ)
In this section, we define gamma (p, s, k)-function p Γ s,k (ξ) in terms of limit function and give its integral representation. Also, we define beta (p, s, k)-function p B s,k (ξ, y) and its integral representation. Furthermore, we prove some identities of the Pochhammer (p, s, k)-symbol p [ξ] n,k,s and the gamma (p, s, k)-function p Γ s,k (ξ).

Theorem 3. e relation between three parameters, two parameters, and the classical Pochhammer's symbol is given by
Proof. Using (14) and (9), we get the desired result. □ Theorem 4. e relation between gamma (p, s, k)-function, gamma (p, k)-function, gamma k-function, and classic gamma function is given by Proof. Using (21) and (8)

Definition and Convergence Condition of the
Mittag-Leffler (p, s, k)-Function p E 9,s k,θ,ϑ (z) In this section, we define a new generalization of the Mittag-Leffler (p, s, k)-function. Also, we check the convergence of the Mittag-Leffler (p, s, k)-function.
where p [ϱ] n,k,s is Pochhammer (p, s, k)-symbol defined in (14), and p Γ s,k (ξ) is defined in (21). e recurrence relation of gamma (p, s, k)-function p Γ s,k (ξ) given in [23] is Now, some characteristics of the Mittag-Leffler (p, s, k)-function are presented. We show that the M-L (p, s, k)-function is an entire function. Also, its order and type are given.

Theorem 6.
e Mittag-Leffler (p, s, k)-function, defined in (38), is an entire function of order ρ and type σ given by Proof. Let R denotes the radius of convergence of the Mittag-Leffler (p, s, k)-function. By considering the properties (5) and (8) and using the asymptotic expansions for the gamma function [1] and the asymptotic Stirling's formula, we have · (|arg(z)| < π; |z| ⟶ ∞).

(41)
In particular, n! � (2πn) 1/2 n n e − n 1 + O n − 1 (n ∈ N; n ⟶ ∞), (42) and the following quotient expansion of two gamma functions at infinity is given as Series (38) can be written in the following forms: since In view of the properties (35) and (36) and using (III.9) of eorem (1) in [22], we get us, the Mittag-Leffler (p, s, k)-function is an entire function.
To obtain the order ρ and the type σ, we apply the following definitions of ρ and σ, respectively: e ρ σ � lim n⟶∞ sup n c n ρ/n .
in the above equation, we get the desired result (53).