Advances in Mathematical Physics
Volume 2020 (2020), Article ID 5792853, 8 pages
https://doi.org/10.1155/2020/5792853

Research Article

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

Gauhar Rahman,¹ Abdus Saboor,² Zunaira Anjum,² Kottakkaran Sooppy Nisar,³ and Thabet Abdeljawad4,5,6

¹Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal 18000, Upper Dir, Pakistan
²Institute of Numerical Sciences, Kohat University of Science & Technology, Kohat, Pakistan, 26000
³Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawser 11991, Saudi Arabia
⁴Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia
⁵Department of Medical Research, China Medical University, Taichung 40402, Taiwan
⁶Department of Computer Science and Information Engineering, Asia University, Taichung 40402, Taiwan

Correspondence should be addressed to Thabet Abdeljawad; tabdeljawad@psu.edu.sa

Received 27 June 2020; Revised 17 August 2020; Accepted 29 August 2020; Published 22 September 2020

Academic Editor: Soheil Salahshour

Copyright © 2020 Gauhar Rahman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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