- Research
- Open Access
- Published:
Some properties of the Mittag-Leffler functions and their relation with the Wright functions
Advances in Difference Equations volume 2012, Article number: 181 (2012)
Abstract
This paper is a short description of our recent results on an important class of the so-called Mittag-Leffler functions, which became important as solutions of fractional order differential and integral equations, control systems and refined mathematical models of various physical, chemical, economical, management and bioengineering phenomena. We have studied the Mittag-Leffler functions as their typical representatives, including many interesting special cases that have already proven their usefulness in fractional calculus and its applications. We obtained a number of useful relationships between the Mittag-Leffler functions and the Wright functions. The Wright function plays an important role in the solution of a linear partial differential equation. The Wright function, which we denote by , is so named in honor of Wright who introduced and investigated this function in a series of notes starting from 1933 in the framework of the asymptotic theory of partitions.
MSC:33E12.
1 Introduction
1.1 The Mittag-Leffler function
The Mittag-Leffler function is an important function that finds widespread use in the world of fractional calculus. Just as the exponential naturally arises out of the solution to integer order differential equations, the Mittag-Leffler function plays an analogous role in the solution of non-integer order differential equations. In fact, the exponential function itself is a very special form, one of an infinite set of these seemingly ubiquitous functions. The standard definition of Mittag-Leffler [1] is given as follows:
A two-parameter function of the M-L (Mittag-Leffler) type is defined by the series expansion [2]
The M-L function provides a simple generalization of the exponential function because of the substitution of with . Particular cases of (2) recover elementary functions are recovered
and
where erf (erfc) denotes the (complementary) error function defined as [3]
By means of the series representation, a generalization of (1) and (2) is introduced by Prabhakar [4] as
where is the Pochhammer symbol [5] given by
Note that
Some new properties of the Mittag-Leffler function, including a definite integral and recurrence relation, were investigated in [6, 7].
1.2 The Wright function
The Wright function plays an important role in the solution of a linear partial differential equation. The Wright function, which we denote by , is so named in honor of Wright, who introduced and investigated this function in a series of notes starting from 1933 in the framework of the asymptotic theory of partitions. This function was introduced that related Mittag-Leffler [8–11]. We obtained a number of useful relationships between the Mittag-Leffler functions and the Wright functions.
1.2.1 Definition
The Wright function is defined by the series representation, convergent in the whole z-complex plane [12]
1.2.2 The integral representation of the Wright function
where Ha denotes the Hankel path. To prove the Hankel path, let us write the integrated function in the form of a power series in z and perform term-by-term integration using the integral representation formula for the reciprocal gamma function
In fact,
1.2.3 The Laplace transform of the Wright function
We recall that the Mittag-Leffler function plays fundamental roles in applications of fractional calculus like fractional relaxation and fractional oscillation [13–16]. Kiryakova introduced and studied the multi-index Mittag-Leffler functions as their typical representatives, including many interesting special cases that have already proven their usefulness in FC and its applications [17]. Srivatava and Tomovski introduced and investigated the fractional calculus with an integral operator which contains the following family of generalized Mittag-Leffler functions [18]. Haubold, Mathaian and Saxena studied the Mittag-Leffler functions and their applications [19]. There is an interesting link between the Wright function and the Mittag-Leffler function. We now point out that the Wright function is related to the Mittag-Leffler function through the following Laplace transform pair:

2 Some properties of the Mittag-Leffler functions
Theorem 1 (Derivative of the Mittag-Leffler function)
If , , , , and , then
Proof Using definition (3), we have that
The Wright function is expressed with help of the Mittag-Leffler function:
□
Theorem 2 (Integration of the Mittag-Leffler function)
If , , , , and , then
Proof According to the Mittag-Leffler function, we have
Integrating both sides gives
Relation with the Wright functions is as follows:
□
Theorem 3 Let , , , , , , then
Proof By definition (3), we have that
Relation with the Wright functions is as follows:
□
Theorem 4 For , , , , , . ‘Note that’

Proof We have


Equation (6) can be written as follows:
We find from equation (7) that
or

We now say each summation on the right-hand side of equation (9) is as follows:
or

We find from equation (11) that
Using equations (9), (11), and (12), we get
Relation with the Wright functions is as follows:
□
Theorem 5 If , , , , , , then
Proof We have from (13)
For , we obtain
Relation with the Wright functions is as follows:
□
References
Mittag-Leffler G:Sur la nouvelle fontion . Comptes Rendus Hebdomadaires Des Seances Del Academie Des Sciences, Paris 2 1903, 137: 554–558.
Wiman A: Über den fundamental Satz in der Theorie der Funktionen. Acta Math. 1905, 29: 191–201. 10.1007/BF02403202
Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
Prabhakar TR: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 1971, 19: 7–15.
Rainville ED: Special Functions. Macmillan, New York; 1960.
Gupta IS, Debnath L: Some properties of the Mittag-Leffer functions. Integral Transforms Spec. Funct. 2007, 18(5):329–336. 10.1080/10652460601090216
Peng J, Li K: A note on property of the Mittag-Leffler function. J. Math. Anal. Appl. 2010, 370(2):635–638. 10.1016/j.jmaa.2010.04.031
Wright EM: On the coefficients of power series having exponential singularities. J. Lond. Math. Soc. 1933, 8: 71–79. 10.1112/jlms/s1-8.1.71
Wright EM: The asymptotic expansion of the generalized Bessel function. Proc. Lond. Math. Soc. 1935, 38: 257–270. 10.1112/plms/s2-38.1.257
Wright EM: The asymptotic expansion of the generalized hypergeometric function. J. Lond. Math. Soc. 1935, 10: 287–293.
Wright EM: The generalized Bessel function of order greater than one. Q. J. Math., Oxford Ser. 1940, 11: 36–48. 10.1093/qmath/os-11.1.36
Erdelyi A, Magnus W, Oberhettinger F, Tricomi FG 3. In Higher Transcendental Functions. McGraw-Hill, New York; 1954. chapter 18
Mainardi F, Mura A, Pagnini G: The M -Wright function in time-fractional diffusion processes: a tutorial survey. Int. J. Differ. Equ. 2010. doi:10.1155/2010/104505
Mainardi F, Gorenflo R: Time-fractional derivatives in relaxation processes: a tutorial survey. Fract. Calc. Appl. Anal. 2007, 10: 269–308.
Achar BNN, Hanneken JW, Clarke T: Damping characteristics of a fractional oscillator. Physica A 2004, 339: 311–319. 10.1016/j.physa.2004.03.030
Mainardi F: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London; 2010.
Kiryakova V: The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus. Comput. Math. Appl. 2010, 59: 1885–1895. 10.1016/j.camwa.2009.08.025
Srivastava HM, Tomovski Ž: Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 2009, 211: 198–210. 10.1016/j.amc.2009.01.055
Haubold HJ, Mathaiand AM, Saxena RK: Mittag-Leffler functions and their applications. J. Appl. Math. 2011., 2011: Article ID 298628. doi:10.1155/2011/298628
Acknowledgements
This work was supported by the scientific and technological research council of Turkey (TUBITAK).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MK defined the research theme. MK designed theorems and proofs, analyzed the data, interpreted the results and wrote the paper. MB participated drafted the manuscript. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Kurulay, M., Bayram, M. Some properties of the Mittag-Leffler functions and their relation with the Wright functions. Adv Differ Equ 2012, 181 (2012). https://doi.org/10.1186/1687-1847-2012-181
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2012-181
Keywords
- Mittag-Leffler functions
- the Wright functions