An Extension of the Mittag-Leffler Function and Its Associated Properties

Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal 18000, Upper Dir, Pakistan Institute of Numerical Sciences, Kohat University of Science & Technology, Kohat, Pakistan 26000 Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawser 11991, Saudi Arabia Department of Mathematics and General Sciences, 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


Introduction
The well-known ML (Mittag-Leffler) function with one parameter is defined by The generalization of (1) with two parameters defined by was presented and contemplated by Mittag-Leffler [1][2][3][4][5][6][7] and other researchers. In [8], the generalization of (1) was given by Shukla and Prajapati [9] defined the following generalization of the ML function by Rahman et al. [10] defined the following extension of generalized ML function by q > 0, and B p (x,y) is the extension of beta function (see [11]). Moreover, the generalization of ML function (3) was presented by [12] as follows: where ðκ ; pÞ l is the Pochhammer symbol which is defined as The researchers studied these extensions (6) and (7) and investigated their further extensions and associated properties and applications. (The readers may consult [13][14][15][16].) Recently, Srivastava et al. [17] have presented and concentrated in a fairly productive way the following extension of the generalized hypergeometric function: where δ j ∈ ℂ for j = 1, 2, ⋯, s, ζ k ∈ ℂ for k = 1, 2, ⋯, t, and ζ k ≠ 0, −1,−2, ⋯, and where ðμ ; ω, aÞ η is the extension of the generalized Pochhammer symbol defined by [23]: The integral representation of ðμ ; ω, aÞ η is explained by where K a ð·Þ is the modified Bessel function of order a. Clearly, when a = 0 in (10), at that point, by utilizing the way that K 1/2 ðtÞ = ffiffiffiffiffiffiffiffi ffi π/2t p e −t , it will lead to formula [(31)]: Specifically, the relating extensions of the confluent hypergeometric function 1F 1 and the Gauss hypergeometric function 2 F 1 are given by The extension of generalized hypergeometric function r F s of r numerator and s denominator parameters was investigated by [18]. Recently, the researchers defined various extensions of special functions and their associated properties and applications in the diverse field. (The interested readers may consult [19][20][21][22].) In [23][24][25], the authors introduced an extension of fractional derivative operators based on the extended beta functions. Next, motivated by the above such extensions of special functions, we define an extension of ML function (6) in terms of the generalized Pochhammer symbol (9) and investigate its certain variations.

Extension of ML Function
We present an extension of the generalized ML function in (6) regarding the extended Pochhammer symbol in (9) as follows: given that the series on the right hand side converges. Clearly, it diminishes to the extended generalized ML function (6) for ν = 0. The special case for a = 1 in (14) can be communicated regarding extended confluent hypergeometric function (13) as follows: 3. Basic Properties of ε κ a,b;p,ν ðz 1 Þ In this section, we present certain basic properties and integral representations of the extended generalized ML function ε κ a,b;p,ν ðz 1 Þ in (14). (14), the following relation holds true:

Advances in Mathematical Physics
Specifically, we have Proof. From (14), we have Equation (17) can be obtained from (16) when we put ν = 0.

Representation of ε κ a,b;p,ν ðz 1 Þ in terms of Generalized Hypergeometric Function
Here, we establish the representation of ε κ a,b;p,ν ðz 1 Þ (14) in terms of generalized hypergeometric function as follows.  (14) for a ∈ N can be represented in the form of generalized hypergeometric function as given by where q ∈ ℕ and Δðq ; bÞ is an array p f q parameters b/q, ðb + 1Þ/q, ⋯, ðb + q − 1Þ/q.
Proof. Taking a = q ∈ ℕ in (14) and utilizing the well-known multiplication formula for the gamma function, we have

Integral Transformation of ε κ a,b,p,ν ðz 1 Þ
Here, we present various integral representations of the function ε κ a,b;p,ν ðz 1 Þ in (14) such as the Mellin, the Euler-beta, and the Laplace transformations.

Mellin Transform.
The well-known Millen transform [26] of integrable function f ðz 1 Þ with index s is defined by if the improper integral in (31) exists.