Abstract
Inspired by certain fascinating ongoing extensions of the special functions such as an extension of the Pochhammer symbol and generalized hypergeometric function, we present a new extension of the generalized Mittag-Leffler (ML) function in terms of the generalized Pochhammer symbol. We then deliberately find certain various properties and integral transformations of the said function . Some particular cases and outcomes of the main results are also established.
1. 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 [1–7] 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 where , and Bp(x,y) is the extension of beta function (see [11]).
Moreover, the generalization of ML function (3) was presented by [12] as follows: where 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 [13–16].) Recently, Srivastava et al. [17] have presented and concentrated in a fairly productive way the following extension of the generalized hypergeometric function: where for , for , and , and where is the extension of the generalized Pochhammer symbol defined by [23]:
The integral representation of is explained by where is the modified Bessel function of order . Clearly, when in (10), at that point, by utilizing the way that , it will lead to formula [(31)]:
Specifically, the relating extensions of the confluent hypergeometric function and the Gauss hypergeometric function are given by
The extension of generalized hypergeometric function of numerator and 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 [19–22].) In [23–25], 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.
2. 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 . The special case for in (14) can be communicated regarding extended confluent hypergeometric function (13) as follows:
3. Basic Properties of
In this section, we present certain basic properties and integral representations of the extended generalized ML function in (14).
Theorem 1. For the function in (14), the following relation holds true:Specifically, we have
Proof. From (14), we have Equation (17) can be obtained from (16) when we put .
Theorem 2. For the function in (14), the following higher order differentiation formulas hold true:
where .
Specifically, we have
Proof. Operating term wise differentiation times on (14), we get In a similar manner from (20), we get Moreover, putting in (20) gives (21). For the special case of (19), (20), and (21), when we put , we get (22), (23), and (24), respectively.
Corollary 3. The following integral representations for ML function (14) hold true:
where
Specifically, we have
4. Representation of in terms of Generalized Hypergeometric Function
Here, we establish the representation of (14) in terms of generalized hypergeometric function as follows.
Theorem 4. The function defined in (14) for can be represented in the form of generalized hypergeometric function as given by where and is an array parameters .
Proof. Taking in (14) and utilizing the well-known multiplication formula for the gamma function, we have
5. Integral Transformation of
Here, we present various integral representations of the function in (14) such as the Mellin, the Euler-beta, and the Laplace transformations.
5.1. Mellin Transform
The well-known Millen transform [26] of integrable function with index is defined by if the improper integral in (31) exists.
Theorem 5. For the function in (14), the following Mellin transform exists:
Proof. By (32), we get from (14) Interchanging the order of summation and integration, we get Now, by utilizing the well-known result ([11], Equation 4.105), Therefore, we have
Corollary 6. The below integral transform exists: where is the Wright hypergeometric function (see [27, 28]).
5.2. Euler-Beta Transform
In [26], the well-known Euler-beta transform of the function is defined by
Theorem 7. For the function in (14), the following Euler-beta transform holds:
Proof. By using (40) and (14), we obtain
Corollary 8. By putting and in (33) and then using (14), we have Similarly, we have In general, we get
5.3. Laplace Transform
The well-known Laplace transform [26] of is defined by
Theorem 9. For the function in (14), the following Laplace transform holds:
Proof. By using (45) and from (14), we have
Corollary 10. By setting and in (46), we obtain
5.4. Whittaker Transformation
To determine the Whittaker transforms, we use the following formula:
Theorem 11. For the function in (14), the following Whittaker transform holds:
Proof. By the definition of Whittaker transform, we have from (14) By putting and then using the definition of Whittaker transforms, we get
6. Conclusion
In our current investigation, we presented an extension of the generalized ML function in (14) by utilizing an extension of Pochhammer symbol defined in (9). Further, we have investigated several basic properties of the newly defined function . The special cases of the main result for can be found in the work of [12]. Thus, the results introduced in this present article are new and an extension of the relating outcomes in the existing literature (see, e.g., [29–31]). The newly defined ML function presented in this article will be applicable in different fields of applied sciences.
Data Availability
No data was used for this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All the authors contributed equally.
Acknowledgments
The author Thabet Abdeljawad would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.