The New Mittag-Leffler Function and Its Applications

Abstract

In this paper, we investigate some properties of the Pochhammer -symbol and gamma -function . We then prove several identities for newly defined symbol and the function . The integral representations for the gamma -function and beta -function are presented. Also, we define a new Mittag-Leffler -function and study its analytic properties and its transforms.

1. Introduction

The 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. This field also has wide applications in both pure mathematics and applied mathematics. The interested readers may consult the literature [1–4].

The 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. The Mittag-Leffler functions have to gain more recognition due to its wide applications in diverse fields. We suggest the readers to review the literature [5–16] for more details.

Throughout this article, let , and be the sets of complex numbers, positive real numbers, negative integers, and natural numbers, respectively.

2. Preliminaries

This section contains some basic definitions and mathematical preliminaries. We begin with the well-known Mittag-Leffler function.

In [17], Gosta Mittag-Leffler introduced the following Mittag-Leffler function which is defined by

In [18], a generalization of Mittag-Leffler function (1) is given by

In [6], Prabhakar proposed the following three parameters of Mittag-Leffler function, which is defined bywhere is the well-known Pochhammer’s symbol defined as follows (see [19]). For ,

Definition 1. In [20], Gehlot introduced the two parameters Pochhammer’s symbol and two parameters of gamma -function.
Let , with , and . Then, the Pochhammer -symbol is defined asand the gamma -function is defined by

Definition 2. The gamma -function is also represented by the following forms:Also, the relationship between the Pochhammer -symbol, Pochhammer -symbol, and the classical Pochhammer’s symbol is represented by [4]The gamma -function satisfies the following relation:where

Definition 3. In [21], the Mittag-Leffler -function is defined bywhere , , , , , , and .
Recently, Cerutti et al. [22] introduced the following Mittag-Leffler -function is defined as follows.

Definition 4. For ,where is the Pochhammer -symbol given by (5) and is gamma -function given by (6).
In [23], Pochhammer -symbol is defined as follows.

Definition 5. For , , and ,whereThe following identities are satisfied:(1)(2)(3)(4)

Definition 6. In [23], the gamma -function in term of -series is given byThe relationship between and is given by [23]The beta -function is defined by

Definition 7. The beta -function is represented by the following integral:

Definition 8. The well-known Laplace transform of piecewise continuous function is defined by

3. Applications and Properties of the Pochhammer -Symbol and Gamma -Function

In this section, we define gamma -function in terms of limit function and give its integral representation. Also, we define beta -function and its integral representation. Furthermore, we prove some identities of the Pochhammer -symbol and the gamma -function .

Definition 9. Suppose that , , , with , and . Then, the gamma -function is given as

Theorem 1. Suppose that , with , and . Then, the following identities hold:

Proof. The properties (22), (23), and (24), respectively, follow from definition (11) and equations (2.8), (2.9), and (2.10) of [16]. The properties (25), (26), and (27), respectively, follow by using equation (21) and (2.11), (2.12), and (2.13) of [16].

Theorem 2. Let , with , and . Then, the following integral representation of gamma -function is defined by

Proof. Consider the right hand side of (28). By applying ([8], page 2) Tannery theorem and using (21), we haveLet , be given byAfter integrating by parts, we obtainAlso,Therefore,

Definition 10. Let , , and . Then, the integral representation of is given bywhere .

Theorem 3. The 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. The relation between gamma -function, gamma -function, gamma -function, and classic gamma function is given by

Proof. Using (21) and (8), we get the desired result.

Theorem 5. Given , and , the recurrence relation for Pochhammer -symbol is given by

Proof. Using the definition of Pochhammer -symbol, we get the desired result.

4. Definition and Convergence Condition of the Mittag-Leffler -Function

In this section, we define a new generalization of the Mittag-Leffler -function. Also, we check the convergence of the Mittag-Leffler -function.

Definition 11. Suppose that . Then, Mittag-Leffler -function is defined bywhere is Pochhammer -symbol defined in (14), and is defined in (21).The recurrence relation of gamma -function given in [23] isNow, some characteristics of the Mittag-Leffler -function are presented. We show that the M-L -function is an entire function. Also, its order and type are given.

Theorem 6. The Mittag-Leffler -function, defined in (38), is an entire function of order and type given by

Proof. Let denotes the radius of convergence of the Mittag-Leffler -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 haveIn particular,and the following quotient expansion of two gamma functions at infinity is given asSeries (38) can be written in the following forms:sinceIn view of the properties (35) and (36) and using of Theorem (1) in [22], we getThus, the Mittag-Leffler -function is an entire function.
To obtain the order and the type , we apply the following definitions of and , respectively:ConsiderBy using Theorem (1) equation ([22]) and definition of (47), we getSimilarly, by putting the value of in the definition of (48) and simplify as the same in Theorem (1) in equation ([22]), we obtain

5. Applications and Properties of the Mittag-Leffler -Function

Some basic properties of Mittag-Leffler -function are presented in this section.

Theorem 7. Suppose that with and , then(1) (2) 

Proof. (1)Taking L.H.S of (42), By using in the above equation, we get the desired result (52).(2)Taking L.H.S of (43), By using in the above equation, we get the desired result (53).

Theorem 8. Suppose that and and with and , then

Proof. Consider the L.H.S.By changing integration and summation orders in the above equation, we haveBy using the recurrence relation of and and equation in [22], we getBy using the above result in equation (60), we have

6. The Euler-Beta Transform of the Mittag-Leffler -Function

The well-known Euler-beta transform is defined bywhere and .

Next, we define the beta transform of newly defined M-L function.

Theorem 9. Suppose that and with , , and , then

Proof. After interchanging the integration and summation orders of the above equation and using the relation between the gamma -function and the classical gamma function given by (3.4), we haveAfter simplification, we obtain the desired result:

Remark 1. (i)If , then equation (64) coincides with the result of [22], sec (ii)If , then equation (64) coincides with the formula in [21](iii)If , then equation (64) coincides with the formula in [24]

7. The Laplace Transform of the Mittag-Leffler -Function

Theorem 10. Let and with , and . Then, the Laplace transform of is given by

Proof. Applying the Laplace transform on the left hand side of (68), the Laplace transform of the potential function (see [1], equation (1.4.58)) and the generalized binomial formula is given byWe haveNow, using the relation between and , we haveWriting the series in compact form, we have

Remark 2. (a)If in the above theorem, then the result coincides with the formula of [22](b)If in Theorem 10, thenwhich coincides with formula (11.13) of [25].

8. Conclusion

In this paper, we established a new extension of the Mittag-Leffler function and investigated some of its properties. We concluded as follows:(1)If , then we get the results of the Mittag-Leffler -function defined in [22](2)If , then we get the results of the Mittag-Leffler -function defined in [21](3)If , then we get the results of the Mittag-Leffler defined in [6](4)If , then we get the results of the Mittag-Leffler defined in [18](5)If , then we get the results of the Mittag-Leffler -function defined in [17](6)If , then we get the exponential function

Recently, the Mittag-Leffler function is used to construct the fractional operators with nonsingular kernels [26, 27]. In [28, 29], the authors introduced the generalized fractional integrals and differential operators, which contain the Mittag-Leffler -function in the kernels, and proved their various properties. Recently, Samraiz et al. [30] introduced the Hilfer Prabhakar -fractional derivative by using the Mittag-Leffler -function. They discussed its various properties and the generalized Laplace transform of the said operator. They also discussed the applications of Hilfer Prabhakar’s -derivative in mathematical physics. In this study, we defined further generalization of the Mittag-Leffler and proved its various basic properties. Hence, it would be of great interest that the Mittag-Leffler function studied in this article will be utilized to generalize such classes of fractional and differential operators.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

U.M. gave the main idea, and all other authors contributed equally to improve the final manuscript.

Acknowledgments

The author T. 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).