#### Abstract

Motivated essentially by the success of the applications of the Mittag-Leffler functions in many areas of science and engineering, the authors present, in a unified manner, a detailed account or rather a brief survey of the Mittag-Leffler function, generalized Mittag-Leffler functions, Mittag-Leffler type functions, and their interesting and useful properties. Applications of G. M. Mittag-Leffler functions in certain areas of physical and applied sciences are also demonstrated. During the last two decades this function has come into prominence after about nine decades of its discovery by a Swedish Mathematician Mittag-Leffler, due to the vast potential of its applications in solving the problems of physical, biological, engineering, and earth sciences, and so forth. In this survey paper, nearly all types of Mittag-Leffler type functions existing in the literature are presented. An attempt is made to present nearly an exhaustive list of references concerning the Mittag-Leffler functions to make the reader familiar with the present trend of research in Mittag-Leffler type functions and their applications.

#### 1. Introduction

The special function and its general form with being the set of complex numbers are called Mittag-Leffler functions [1, Section 18.1]. The former was introduced by Mittag-Leffler [2, 3], in connection with his method of summation of some divergent series. In his papers [24], he investigated certain properties of this function. Function defined by (1.2) first appeared in the work of Wiman [5, 6]. The function (1.2) is studied, among others, by Wiman [5, 6], Agarwal [7], Humbert [8], and Humbert and Agarwal [9] and others. The main properties of these functions are given in the book by Erdélyi et al. [1, Section 18.1], and a more comprehensive and a detailed account of Mittag-Leffler functions is presented in Dzherbashyan [10, Chapter 2]. In particular, functions (1.1) and (1.2) are entire functions of order and type ; see, for example, [1, page 118].

The Mittag-Leffler function arises naturally in the solution of fractional order integral equations or fractional order differential equations, and especially in the investigations of the fractional generalization of the kinetic equation, random walks, Lévy flights, superdiffusive transport and in the study of complex systems. The ordinary and generalized Mittag-Leffler functions interpolate between a purely exponential law and power-law-like behavior of phenomena governed by ordinary kinetic equations and their fractional counterparts, see Lang [11, 12], Hilfer [13, 14], and Saxena [15].

The Mittag-Leffler function is not given in the tables of Laplace transforms, where it naturally occurs in the derivation of the inverse Laplace transform of the functions of the type , where is the Laplace transform parameter and and are constants. This function also occurs in the solution of certain boundary value problems involving fractional integrodifferential equations of Volterra type [16]. During the various developments of fractional calculus in the last four decades this function has gained importance and popularity on account of its vast applications in the fields of science and engineering. Hille and Tamarkin [17] have presented a solution of the Abel-Volterra type equation in terms of Mittag-Leffler function. During the last 15 years the interest in Mittag-Leffler function and Mittag-Leffler type functions is considerably increased among engineers and scientists due to their vast potential of applications in several applied problems, such as fluid flow, rheology, diffusive transport akin to diffusion, electric networks, probability, and statistical distribution theory. For a detailed account of various properties, generalizations, and application of this function, the reader may refer to earlier important works of Blair [18], Torvik and Bagley [19], Caputo and Mainardi [20], Dzherbashyan [10], Gorenflo and Vessella [21], Gorenflo and Rutman [22], Kilbas and Saigo [23], Gorenflo and Luchko [24], Gorenflo and Mainardi [25, 26], Mainardi and Gorenflo [27, 28], Gorenflo et al. [29], Gorenflo et al. [30], Luchko [31], Luchko and Srivastava [32], Kilbas et al. [33, 34], Saxena and Saigo [35], Kiryakova [36, 37], Saxena et al. [38], Saxena et al. [3943], Saxena and Kalla [44], Mathai et al. [45], Haubold and Mathai [46], Haubold et al. [47], Srivastava and Saxena [48], and others.

This paper is organized as follows: Section 2 deals with special cases of . Functional relations of Mittag-Leffler functions are presented in Section 3. Section 4 gives the basic properties. Section 5 is devoted to the derivation of recurrence relations for Mittag-Leffler functions. In Section 6, asymptotic expansions of the Mittag-Leffler functions are given. Integral representations of Mittag-Leffler functions are given in Section 7. Section 8 deals with the -function and its special cases. The Melllin-Barnes integrals for the Mittag-Leffler functions are established in Section 9. Relations of Mittag-Leffler functions with Riemann-Liouville fractional calculus operators are derived in Section 10. Generalized Mittag-Leffler functions and some of their properties are given in Section 11. Laplace transform, Fourier transform, and fractional integrals and derivatives are discussed in Section 12. Section 13 is devoted to the application of Mittag-Leffler function in fractional kinetic equations. In Section 14, time-fractional diffusion equation is solved. Solution of space-fractional diffusion equation is discussed in Section 15. In Section 16, solution of a fractional reaction-diffusion equation is investigated in terms of the -function. Section 17 is devoted to the application of generalized Mittag-Leffler functions in nonlinear waves. Recent generalizations of Mittag-Leffler functions are discussed in Section 18.

#### 2. Some Special Cases

We begin our study by giving the special cases of the Mittag-Leffler function .(i)(ii)(iii)(iv)(v)(vi)(vii) where erfc denotes the complimentary error function and the error function is defines as For half-integer the function can be written explicitly as(viii)(ix)

#### 3. Functional Relations for the Mittag-Leffler Functions

In this section, we discuss the Mittag-Leffler functions of rational order , with relatively prime. The differential and other properties of these functions are described in Erdélyi et al. [1] and Dzherbashyan [10].

Theorem 3.1. The following results hold: where denotes the incomplete gamma function, defined by

In order to establish the above formulas, we observe that (3.1) and (3.2) readily follow from definition (1.2). For proving formula (3.3), we recall the identity By virtue of the results (1.1) and (3.6), we find that which can be written as and result (3.3) now follows by taking . To prove relation (3.4), we set in (3.1) and multiply it by to obtain On integrating both sides of the above equation with respect to and using the definition of incomplete gamma function (3.5), we obtain the desired result (3.4). An interesting case of (3.8) is given by

#### 4. Basic Properties

This section is based on the paper of Berberan-Santos [49]. From (1.1) and (1.2) it is not difficult to prove that It is shown in Berberan-Santos [49, page 631] that the following three equations can be used for the direct inversion of a function to obtain its inverse : With help of the results (4.2) and (4.4), it yields the following formula for the inverse Laplace transform of the function : In particular, the following interesting results can be derived from the above result: Another integral representation of in terms of the Lévy one-sided stable distribution was given by Pollard [50] in the form The inverse Laplace transform of , denoted by with , is obtained as where is the one-sided Lévy probability density function. From Berberan-Santos [49, page 432] we have Expanding the above equation in a power series, it gives with The Laplace transform of (4.11) is the asymptotic expansion of as

#### 5. Recurrence Relations

By virtue of definition (1.2), the following relations are obtained in the following form:

Theorem 5.1. One has

The above formulae are useful in computing the derivative of the Mittag-Leffler function . The following theorem has been established by Saxena [15].

Theorem 5.2. If , and then there holds the formula

Proof. We have from the right side of (5.2) Put or . Then, For we obtain the following corollaries.Corollary 5.3. If , then there holds the formula Corollary 5.4. If , then there holds the formula Corollary 5.5. If , then there holds the formula

Remark 5.6. For a generalization of result (5.2), see Saxena et al. [38].

#### 6. Asymptotic Expansions

The asymptotic behavior of Mittag-Leffler functions plays a very important role in the interpretation of the solution of various problems of physics connected with fractional reaction, fractional relaxation, fractional diffusion, and fractional reaction-diffusion, and so forth, in complex systems. The asymptotic expansion of is based on the integral representation of the Mittag-Leffler function in the form where the path of integration is a loop starting and ending at and encircling the circular disk in the positive sense, on . The integrand has a branch point at . The complex -plane is cut along the negative real axis and in the cut plane the integrand is single-valued the principal branch of is taken in the cut plane. Equation (6.1) can be proved by expanding the integrand in powers of and integrating term by term by making use of the well-known Hankel's integral for the reciprocal of the gamma function, namely The integral representation (6.1) can be used to obtain the asymptotic expansion of the Mittag-Leffler function at infinity [1]. Accordingly, the following cases are mentioned.(i)If and is a real number such that then for , there holds the following asymptotic expansion: as , and as , . (ii)When then there holds the following asymptotic expansion: as , and where the first sum is taken over all integers such that The asymptotic expansion of is based on the integral representation of the Mittag-Leffler function in the form which is an extension of (6.1) with the same path. As in the previous case, the Mittag-Leffler function has the following asymptotic estimates.(iii)If and is a real number such that then there holds the following asymptotic expansion: as , and as , . (iv)When then there holds the following asymptotic expansion: as , and where the first sum is taken over all integers such that

#### 7. Integral Representations

In this section several integrals associated with Mittag-Leffler functions are presented, which can be easily established by the application by means of beta and gamma function formulas and other techniques, see Erdélyi et al. [1], Gorenflo et al. [29, 51, 52], where and

Note 1. Equation (7.3) can be employed to compute the numerical coefficients of the leading term of the asymptotic expansion of . Equation (7.4) yields From Berberan-Santos [49] and Gorenflo et al. [29, 51] the following results hold: In particular, the following cases are of importance:

Note 2. Some new properties of the Mittag-Leffler functions are recently obtained by Gupta and Debnath [53].

#### 8. The -Function and Its Special Cases

The -function is defined by means of a Mellin-Barnes type integral in the following manner [54]: where and and an empty product is interpreted as unity, with , , , , ; such that where we employ the usual notations: , , , and being the complex number field. The contour is the infinite contour which separates all the poles of , from all the poles of , . The contour could be or or , where is a loop starting at encircling all the poles of , and ending at . is a loop starting at , encircling all the poles of and ending at . is the infinite semicircle starting at and going to . A detailed and comprehensive account of the -function is available from the monographs of Mathai and Saxena [54], Prudnikov et al. [55] and Kilbas and Saigo [56]. The relation connecting the Wright's function and the -function is given for the first time in the monograph of Mathai and Saxena [54, page 11, equation (1.7.8)] as where is the Wright's generalized hypergeometric function [57, 58]; also see Erdélyi et al. [59, Section 4.1], defined by means of the series representation in the form where , , , , ; ; , The Mellin-Barnes contour integral for the generalized Wright function is given by where the path of integration separates all the poles of at the points , lying to the left and all the poles of , at the points , , lying to the right. If , then the above representation is valid if either of the conditions are satisfied:(i)(ii)

This result was proved by Kilbas et al. [60].

The generalized Wright function includes many special functions besides the Mittag-Leffler functions defined by equations (1.1) and (1.2). It is interesting to observe that for , ; , (8.5) reduces to a generalized hypergeometric function . Thus where , , , , , , or , . Wright [61] introduced a special case of (8.5) in the form which widely occurs in problems of fractional diffusion. It has been shown by Saxena et al. [41], also see Kiryakova [62], that If we further take in (8.12) we find that where , .

Remark 8.1. A series of papers are devoted to the application of the Wright function in partial differential equation of fractional order extending the classical diffusion and wave equations. Mainardi [63] has obtained the result for a fractional diffusion wave equation in terms of the fractional Green function involving the Wright function. The scale-variant solutions of some partial differential equations of fractional order were obtained in terms of special cases of the generalized Wright function by Buckwar and Luchko [64] and Luchko and Gorenflo [65].

#### 9. Mellin-Barnes Integrals for Mittag-Leffler Functions

These integrals can be obtained from identities (8.12) and (8.13).

Lemma 9.1. If , and the following representations are obtained: where the path of integration separates all the poles of at the points from those of at the points , .

On evaluating the residues at the poles of the gamma function we obtain the following analytic continuation formulas for the Mittag-Leffler functions:

#### 10. Relation with Riemann-Liouville Fractional Calculus Operators

In this section, we present the relations of Mittag-Leffler functions with the left- and right-sided operators of Riemann-Liouville fractional calculus, which are defined where means the maximal integer not exceeding and is the fractional part of .

Note 3. The fractional integrals (10.1) and (10.2) are connected by the relation [66, page 118]

Theorem 10.1. Let and then there holds the formulas which by virtue of definitions (1.1) and (1.2) can be written as

Theorem 10.2. Let and then there holds the formulas

Theorem 10.3. Let and then there holds the formulas

Theorem 10.4. Let and then there holds the formula

#### 11. Generalized Mittag-Leffler Type Functions

By means of the series representation a generalization of (1.1) and (1.2) is introduced by Prabhakar [67] as where whenever is defined, . It is an entire function of order and type . It is a special case of Wright's generalized hypergeometric function, Wright [57, 68] as well as the -function [54]. For various properties of this function with applications, see Prabhakar [67]. Some special cases of this function are enumerated below where is the Kummer's confluent hypergeometric function. has the following representations in terms of the Wright's function and -function: where and are, respectively, Wright generalized hypergeometric function and the -function. In the Mellin-Barnes integral representation, and the in the contour is such that , and it is assumed that the poles of and are separated by the contour. The following two theorems are given by Kilbas et al. [34].

Theorem 11.1. If , , , then for the following results hold: In particular,

Theorem 11.2. If , , , , , then,

The proof of (11.7) can be developed with the help of the Laplace transform formula where , , , , , , . For , (11.8) reduces to Generalization of the above two results is given by Saxena [15] where .

Relations connecting the function defined by (11.1) and the Riemann-Liouville fractional integrals and derivatives are given by Saxena and Saigo [35] in the form of nine theorems. Some of the interesting theorems are given below.

Theorem 11.3. Let , , , and . Let be the left-sided operator of Riemann-Liouville fractional integral. Then there holds the formula

Theorem 11.4. Let , , , and . Let be the right-sided operator of Riemann-Liouville fractional integral. Then there holds the formula

Theorem 11.5. Let , , , and . Let be the left-sided operator of Riemann-Liouville fractional derivative. Then there holds the formula

Theorem 11.6. Let , , , and . Let be the right-sided operator of Riemann-Liouville fractional derivative. Then there holds the formula

In a series of papers by Luchko and Yakubovich [69, 70], Luckho and Srivastava [32], Al-Bassam and Luchko [71], Hadid and Luchko [72], Gorenflo and Luchko [24], Gorenflo et al. [73, 74], Luchko and Gorenflo [75], the operational method was developed to solve in closed forms certain classes of differential equations of fractional order and also integral equations. Solutions of the equations and problems considered are obtained in terms of generalized Mittag-Leffler functions. The exact solution of certain differential equation of fractional order is given by Luchko and Srivastava [32] in terms of function (11.1) by using operational method. In other papers, the solutions are established in terms of the following functions of Mittag-Leffler type: if , , and then, For , (11.15) reduces to (11.1). The Mellin-Barnes integral for this function is given by where 0, , , and the contour separates the poles of from those of . , , . The Laplace transform of the function defined by (11.15) is given by where .

Remark 11.7. In a recent paper, Kilbas et al. [33] obtained a closed form solution of a fractional generalization of a free electron equation of the form where , , , , , , , and is the generalized Mittag-Leffler function given by (11.1), and is the unknown function to be determined.

Remark 11.8. The solution of fractional differential equations by the operational methods are also obtained in terms of certain multivariate Mittag-Leffler functions defined below: The multivariate Mittag-Leffler function of complex variables with complex parameters is defined as in terms of the multinomial coefficients

Another generalization of the Mittag-Leffler function (1.2) was introduced by Kilbas and Saigo [23, 76] in terms of a special function of the form where an empty product is to be interpreted as unity; are complex numbers and , , , , and for the above defined function reduces to a constant multiple of the Mittag-Leffler function, namely where and . It is an entire function of of order and type , see Gorenflo et al. [30]. Certain properties of this function associated with Riemann-Liouville fractional integrals and derivatives are obtained and exact solutions of certain integral equations of Abel-Volterra type are derived by their applications [23, 76, 77]. Its recurrence relations, connection with hypergeometric functions and differential formulas are obtained by Gorenflo et al. [30], also see, Gorenflo and Mainardi [25]. In order to present the applications of Mittag-Leffler functions we give definitions of Laplace transform, Fourier transform, Riemann-Liouville fractional calculus operators, Caputo operator and Weyl fractional operators in the next section.

#### 12. Laplace and Fourier Transforms, Fractional Calculus Operators

We will need the definitions of the well-known Laplace and Fourier transforms of a function and their inverses, which are useful in deriving the solution of fractional differential equations governing certain physical problems. The Laplace transform of a function with respect to is defined by where and its inverse transform with respect to is given by The Fourier transform of a function with respect to is defined by The inverse Fourier transform with respect to is given by the formula From Mathai and Saxena [54] and Prudnikov et al. [55, page 355, (2.25.3)] it follows that the Laplace transform of the -function is given by where , , , By virtue of the cancelation law for the -function [54] it can be readily seen that