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The extended Mittag-Leffler function and its properties
Journal of Inequalities and Applications volume 2014, Article number: 85 (2014)
In this paper, we present the extended Mittag-Leffler functions by using the extended Beta functions (Chaudhry et al. in Appl. Math. Comput. 159:589-602, 2004) and obtain some integral representations of them. The Mellin transform of these functions is given in terms of generalized Wright hypergeometric functions. Furthermore, we show that the extended fractional derivative (Özarslan and Özergin in Math. Comput. Model. 52:1825-1833, 2010) of the usual Mittag-Leffler function gives the extended Mittag-Leffler function. Finally, we present some relationships between these functions and the Laguerre polynomials and Whittaker functions.
Fractional differential equations have been an active research area during the past few decades and they occur in many applications of physics and engineering. The Mittag-Leffler function appears as the solution of fractional order differential equations and fractional order integral equations. Some applications of the Mittag-Leffler function are as follows: studies of the kinetic equation, the telegraph equation , random walks, Levy flights, superdiffuse transport, and complex systems. Besides this, the Mittag-Leffler function appears in the solution of certain boundary value problems involving fractional integro-differential equations of Volterra type . It has applications in applied problems, such as fluid flow, rheology, diffusive transport akin to diffusion, electric networks, probability, and statistical distribution theory. Various properties of the Mittag-Leffler functions were presented and surveyed in . Furthermore, a different variant of the Mittag-Leffler function has been investigated in .
Let us start with giving the historical background of the Mittag-Leffler functions. The function ,
was defined and studied by Mittag-Leffler in the year 1903 in [5–7]. It is a direct generalization of the exponential series, since, for , we have the exponential function. The function defined by
gives a generalization of equation (1). This generalization was studied by Wiman in 1905 [8, 9], Agarwal in 1953, and Humbert and Agarwal [10, 11] in 1953. Afterward, Prabhakar  introduced the generalized Mittag-Leffler function by
where with . For , it reduces to the Mittag-Leffler function given in equation (2). Some of the properties of the generalized Mittag-Leffler function such as the Mellin transform, the inverse Mellin transform, and differentiation were given in . On the other hand, monotony of the Mittag-Leffler function was given in .
In this paper, we extend the Mittag-Leffler function in the following way. Since
using the fact that
we extend the Mittag-Leffler function as follows:
where for we have
the extended Euler’s Beta function defined in  (see also ).
The organization of the paper is as follows: In Section 2, we give an integral representations of the extended Mittag-Leffler function in terms of Prabhakar’s Mittag-Leffler function and in terms of known elementary functions. The Mellin transform of the extended Mittag-Leffler function is obtained by means of the generalized Wright hypergeometric function . In Section 3, we obtain fractional derivative representations of the extended Mittag-Leffler function and give some derivative formulas. In Section 4, we obtain the relationship between the extended Mittag-Leffler function and simple Laguerre polynomials and Whittaker’s functions.
2 Some properties of the extended Mittag-Leffler function
We begin with the following theorem, which gives the integral representation of the extended Mittag-Leffler function.
Theorem 1 (Integral representation)
For the extended Mittag-Leffler function, we have
where , , , .
Proof Using equation (5) in equation (4), we get
Interchanging the order of summation and integration in equation (7), which is guaranteed under the assumptions given in the statement of the theorem, we get
Using equation (3) in equation (8), we get the desired result. □
Corollary 2 Note that, taking in Theorem 1, we get
Corollary 3 Taking in the Theorem 1, we get the following integral representation:
Now, using the definition of Prabhakar’s Mittag-Leffler’s function, Bayram and Kurulay obtained the recurrence formula 
Inserting the above recurrence relation into equation (6), we get the following recurrence relation for the extended Mittag-Leffler’s function.
Corollary 4 (Recurrence relation)
For the extended Mittag-Leffler function, we get
where , , , .
In the next theorem, we give the Mellin transform of the extended Mittag-Leffler function in terms of the Wright generalized hypergeometric function. Note that the Wright generalized hypergeometric function is defined by 
where the coefficients () and () are positive real numbers such that
Theorem 5 (Mellin transform)
The Mellin transform of the extended Mittag-Leffler function is given by
where is the Wright generalized hypergeometric function.
Proof Taking the Mellin transform of the extended Mittag-Leffler function, we have
Using equation (6) in equation (13), we get
Interchanging the order of integrals in equation (14), which is valid because of the conditions in the statement of the Theorem 5, we get
Now taking in equation (15) and using the fact that , we get
Using the definition of Prabhakar’s generalized Mittag-Leffler function in equation (16), we get
Interchanging the order of summation and integration, which is valid for , , , , , we get
Using the Beta function in equation (18), we have
Considering that , , and inserting equation (11) into equation (19), we get the result
Corollary 6 Taking in Theorem 5, we get
Corollary 7 Taking the inverse Mellin transform on both sides of equation (12), we get the elegant complex integral representation
3 Derivative properties of the extended Mittag-Leffler function
The classical Riemann-Liouville fractional derivative of order μ is usually defined by
where the integration path is a line from 0 to z in the complex t-plane. For the case (), it is defined by
The extended Riemann-Liouville fractional derivative operator was defined by Özarslan and Özergin as follows.
Definition 8 ()
The extended Riemann-Liouville fractional derivative is defined as
and for ()
where the path of integration is a line from 0 to z in the complex t-plane. For the case , we obtain the classical Riemann-Liouville fractional derivative operator.
We begin by the following theorem.
Theorem 9 Let , , , . Then
Proof Replacing μ by in the definition of the extended fractional derivative operator (20), we get
Taking in equation (21), we get
Comparing this result with equation (6), we get
Whence the result. □
In the following theorem, we give the derivative properties of the extended Mittag-Leffler function.
Theorem 10 For the extended Mittag-Leffler function, we have the following derivative formula:
Proof Taking the derivative with respect to z in equation (6), we get
Again taking the derivative with respect to z in equation (24), we get
Continuing the repetition of this procedure n times, we get the desired result. □
Theorem 11 For the extended Mittag-Leffler function, the following differentiation formula holds:
Proof In equation (23), replace z by and multiply , then taking the z-derivative n times, we get the result. □
Theorem 12 For the extended Mittag-Leffler function, the following differentiation formula holds:
Proof Taking the p-derivative n times in equation (6), we get the result. □
4 Relations between the extended Mittag-Leffler function with Laguerre polynomial and Whittaker function
In this section, we give a representation of the extended Mittag-Leffler function in terms of Laguerre polynomials and Whittaker’s function.
Theorem 13 For the extended Mittag-Leffler function, we have
where , , .
Proof We start by recalling the useful identity used in 
Using equation (26) in equation (6), we get
Now, taking into account the series expansion of Prabhakar’s generalized Mittag-Leffler’s function in equation (27), we have
Interchanging the order of integration and summation in equation (28), which can be done under the assumptions of the theorem, we have
Multiplying both sides of equation (29) by , we get the result. □
In the following theorem, we give the extended Mittag-Leffler function in terms of Whittaker’s function.
Theorem 14 For the extended Mittag-Leffler function we have
Proof Considering the following equality:
and using the generating function of the Laguerre polynomials, we get
Taking equation (30) into account in equation (6), we have
By use of Prabhakar’s generalized Mittag-Leffler function in equation (31), we get
Interchanging the order of summation and integration in equation (32), we get
Finally, using the following integral representation :
in equation (33), we get the result. □
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The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.
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Özarslan, M.A., Yılmaz, B. The extended Mittag-Leffler function and its properties. J Inequal Appl 2014, 85 (2014). https://doi.org/10.1186/1029-242X-2014-85
- extended Beta functions
- fractional derivative
- Mellin transform
- Laguerre polynomials
- Whittaker functions
- Wright generalized hypergeometric functions