Mittag-Leffler Functions and Their Applications

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 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 G.M. Mittag-Leffler, due its vast potential of its applications in solving the problems of physical, biological, engineering and earth sciences etc. 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.

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, super-diffusive 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 (1999aLang ( , 1999b, , . 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 p α (a + bp β ), where p is the Laplace transform parameter and a and b are constants. This function also occurs in the solution of certain boundary value problems involving fractional integro-differential equations of Volterra type (Samko et al., 1993). 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 (1930) 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, statistical distribution theory etc. For a detailed account of various properties, generalizations, and application of this function, the reader may refer to earlier important works of Blair (1974), Bagley and Torvik (1984), Caputo and Mainardi (1971), Dzherbashyan (1966), Gorenflo and Vessella (1991), Gorenflo and Rutman (1994), , , Gorenflo and Mainardi (1994, Gorenflo, Luchko and Rogosin (1997), Gorenflo, Kilbas and Rogosin (1998), Luchko (1999), Luchko and Srivastava (1995), Saxena (2002, 2004), , Kiryakova (2008aKiryakova ( , 2008b, Saxena, Kalla and Kiryakova (2003), Saxena, Mathai and Haubold (2002, 2004a, 2004b, , , Haubold and Mathai (2000), Haubold, Mathai and Saxena (2007), Srivastava and Saxena (2001), and others. This paper is organized as follows: Section 2 deals with special cases of E α (z). 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 H-function and its special cases. The Mellin-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 H-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.

Functional relations for the Mittag-Leffler functions
In this section, we discuss the Mittag-Leffler functions of rational order α = m/n, with m, n ∈ N relatively prime. The differential and other properties of these functions are described in Erdélyi, et al.(1955) and Dzherbashyan (1966).
Theorem 3.1. The following results hold: where γ(a, z) denotes the incomplete gamma function, defined by, which can be written as and the result (3.3) now follows by taking α = m/n. To prove the relation (3.4), we set m = 1 in (3.1) and multiply it by exp(−z) to obtain On integrating both sides of the above equation with respect to z 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 (3.10)

Basic properties
This section is based on the paper of Berberan-Santos (2005). From (1.1) and (1.2) it is not difficult to prove that It is shown in Berberan-Santos (2005, p.631) that the following three equations can be used for the direct inversion of a function I(x) to obtain its inverse H(k): With the help of the results (4.2) and (4.4), it yields the following formula for the inverse Laplace transform H(k) of the function E α (−x).
In particular, the following interesting results can be derived from the above result.
Another integral representation of H α (k) in terms of the Lévy one-sided stable distribution L α (k) was given by Pllard (1948) in the form The inverse Laplace transform of E α (−x β ), denoted by H β α (k) with 0 < α ≤ 1, is obtained as where L α (t) is the one-sided Lévy probability density function. From Berberan-Santos (2005, p.432) we have Expanding the above equation in a power series, it gives (4.14) The Laplace transform of the equation (4.13) is the asymptotic expansion of E α (−x) as

Recurrence relations
By virtue of the definition (1.2), the following relations are obtained in the form of The above formulae are useful in computing the derivative of the Mittag-Leffler function E α,β (z). The following theorem has been established by Saxena (2002): If ℜ(α) > 0, ℜ(β) > 0 and r ∈ N then there holds the formula .
Proof. We have from the right side of (5.5), .

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 etc in complex systems. The asymptotic expansion of E α (z) is based on the integral representation of the Mittag-Leffler function in the form where the path of integration Ω is a loop starting and ending at −∞ and encircling the circular disk |t| ≤ |z| 1 α in the positive sense, | arg t| < π on Ω. The integrand has a branch point at t = 0. The complex t-plane is cut along the negative real axis and in the cut plane the integrand is single-valued, the principal branch of t α is taken in the cut plane. (6.1) can be proved by expanding the integrand in powers of t 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 (Erdélyi, et at., 1955). Accordingly, the following cases are mentioned below: (i): If 0 < α < 2 and µ is a real number such that then for N * ∈ N, N * = 1 there holds the following asymptotic expansion: as |z| → ∞, | arg z| ≤ µ; and as |z| → ∞, µ ≤ | arg z| ≤ π. (ii) : When α ≥ 2 then there holds the following asymptotic expansion: as |z| → ∞, | arg z| ≤ απ 2 , and where the first sum is taken over all integers n such that The asymptotic expansion of E α,β (z) is based on the integral representation of the Mittag-Leffler function E α,β (z) 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 0 < α < 2 and µ is a real number such that then there holds the following asymptotic expansion: as |z| → ∞, | arg z| ≤ µ; and as |z| → ∞, µ ≤ | arg z| ≤ π. (iv): When α ≥ 2 then there holds the following asymptotic expansion: as |z| → ∞, | arg z| ≤ απ 2 and where the first sum is taken over all integers n such that | arg(z) + 2πn| ≤ απ 2 . (6.13)

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. (1955), Gorenflo et al. ( , 2002. Note 7.1. Equation (7.7) can be employed to compute the numerical coefficients of the leading term of the asymptotic expansion of E α (−x). Equation (7.8) yields In particular, the following cases are of importance x 2 + t 2 dt = e x 2 erfc(x). (7.12)

The H-function and its special cases
The H-function is defined by means of a Mellin-Barnes type integral in the following manner (Mathai and Saxena, 1978): where i = (−1) and and an empty product is interpreted as unity; m, n, p, q ∈ N 0 with 0 ≤ n ≤ p, 1 ≤ m ≤ q, A i , B j ∈ R + , a i , b j ∈ C, i = 1, ..., p; j = 1, ..., q such that , (8.4) where p ψ q (z) is the Wright's generalized hypergeometric function (Wright, 1935(Wright, , 1940; also see Erdélyi, et al. (1953, Section 4.1), defined by means of the series representation in the form The Mellin-Barnes contour integral for the generalized Wright function is given by where the path of integration separates all the poles of Γ(s) at the points s = −ν, ν ∈ N 0 lying to the left and all the poles of p j=1 Γ(a j − sA j ), j = 1, ..., p at the points s = (A j + ν j )/A j , ν j ∈ N 0 , j = 1, ..., p lying to the right. If Ω = (γ − i∞, γ + i∞) then the above representation is valid if either of the conditions are satisfied: This result was proved by Kilbas, Saigo and Trujillo (2002).
The generalized Wright function includes many special functions besides the Mittag-Leffler functions defined by the equations (1.1) and (1.2). It is interesting to observe that for A i = B j = 1, i = 1, ..., p; j = 1, ..., q, (8.5) reduces to a generalized hypergeometric function p F q (z). Thus which widely occurs in problems of fractional diffusion. It has been shown by Saxena, Mathai and Haubold (2004a), also see Kiryakova (1994), that .
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 (1997) 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 (1998) and Luchko and Gorenflo (1998).

Mellin-Barnes integrals for Mittag-Leffler functions
These integrals can be obtained from the identities (8.9) and (8.10).
On evaluating the residues at the poles of the gamma function Γ(1 − s) we obtain the following analytic continuation formulas for the Mittag-Leffler functions: and .

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 below.
where [α] means the maximal integer not exceeding α and {α} is the fractional part of α.

Generalized Mittag-Leffler type functions
By means of the series representation a generalization of (1.1) and (1.2) is introduced by Prabhakar (1971) as It is a special case of Wright's generalized hypergeometric function, Wright (1934Wright ( ,1935 as well as the H-function (Mathai and Saxena, 1978). For various properties of this function with applications, see Prabhakar (1971). Some special cases of this function are enumerated below.
where φ(α, β; z) is the Kummer's confluent hypergeometric function. E γ α,β (z) has the following representations in terms of the Wright's function and H-function.
where 1 ψ 1 (·) and H 1,1 1,2 (·) are respectively Wright generalized hypergeometric function and the H-function. In the Mellin-Barnes integral representation, ω = √ −1 and the c in the contour is such that 0 < c < ℜ(γ) and it is assumed that the poles of Γ(s) and Γ(γ − s) are separated by the contour. The following two theorems are given by Kilbas, Saigo and Saxena (2004).
Relations connecting the function defined by (11.1) and the Riemann-Liouville fractional integrals and derivatives are given by  in the form of nine theorems. Some of the interesting theorems are given below.
be the left-sided operator of Riemann-Liouville fractional integral. Then there holds the formula Theorem 11.4. Let α > 0, β > 0, γ > 0 and a ∈ R. Let I α − be the right-sided operator of Riemann-Liouville fractional integral. Then there holds the formula Theorem 11.5. Let α > 0, β > 0, γ > 0 and a ∈ R. Let D α 0+ be the left-sided operator of Riemann-Liouville fractional derivative. Then there holds the formula Theorem 11.6. Let α > 0, β > 0, γ − α + {α} > 1 and a ∈ R. Let D α − be the right-sided operator of Riemann-Liouville fractional derivative. Then there holds the formula In a series of papers by Yakubovich (1990, 1994), Luckho and Srivastava (1995), Al-Bassam and , Hadid and Luchko (1996), , , , 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 (1995) in terms of the 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 z, ρ, β j ∈ C, ℜ(α j ) > 0, j = 1, ..., m and m ∈ N then For m = 1, (11.20) reduces to (11.1). The Mellin-Barnes integral for this function is given by where 0, γ < ℜ(ρ), ℜ(ρ) > 0 and the contour separates the poles of Γ(s) from those of Γ(ρ − s). ℜ(α j ) > 0, j = 1, ..., m, arg(−z) < π. The Laplace transform of the function defined by (11.20) is given by where ℜ(s) > 0.
is the generalized Mittag-Leffler function given by (11.1), and α(τ ) is the unknown function to be determined.
Remark 11.2. 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 n complex variables z 1 , ..., z n with complex parameters a 1 , ..., a n , b ∈ C is defined as (11.25) in terms of the multinomial coefficients Another generalization of the Mittag-Leffler function (1.2) was introduced by Kilbas and Saigo (1995) in terms of a special function of the form where an empty product is to be interpreted as unity; α, β ∈ C are complex numbers and m ∈ R, .. and for m = 1 the above defined function reduces to a constant multiple of the Mittag-Leffler function, namely (11.28) where ℜ(α) > 0 and α(i + β) / ∈ Z − . It is an entire function of z of order [ℜ(α)] −1 and type σ = 1/m, see . 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 (Kilbas andSaigo, 1995, 1996). Its recurrence relations, connection with hypergeometric functions and differential formulas are obtained by Gorenflo, Kilbas and Rogosin (1998), also see, Gorenflo and Mainardi (1996). 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.

Laplace and Fourier transforms, fractional calculus operators
We will need the definitions of the well-known Laplace and Fourier transforms of a function N (x, t) and their inverses, which are useful in deriving the solution of fractional differential equations governing certain physical problems. The Laplace transform of a function N (x, t) with respect to t is defined by where ℜ(s) > 0 and its inverse transform with respect to s is given by The Fourier transform of a function N (x, t) with respect to x is defined by 3) The inverse Fourier transform with respect to k is given by the formula , By virtue of the cancelation law for the H-function (Mathai and Saxena, 1978) it can be readily seen that where In view of the results the cosine transform of the H-function (Mathai and Saxena, 1978, p.49) is given by A j , and k > 0. The definitions of fractional integrals used in the analysis are defined below. The Riemann-Liouville fractional integral of order ν is defined by (Miller and Ross, 1993, p.45) where ℜ(ν) > 0. Following Samko, Kilbas and Marichev (1993, p.37) we define the Riemann-Liouville fractional derivative for α > 0 in the form In certain boundary-value problems arising in the theory of visco-elasticity and in the hereditary solid mechanics the following fractional derivative of order α > 0 is introduced by Caputo (1969) in the form where ∂ m ∂t m f is the m-th partial derivative of the function f (x, t) with respect to t. The Laplace transform of this derivative is given by Podlubny (1999) in the form The above formula is very useful in deriving the solution of differintegral equations of fractional order governing certain physical problems of reaction and diffusion. Making use of the definitions (12.10) and (12.11) it readily follows that for f (t) = t ρ we obtain On taking ρ = 0 in (12.18) we find that From the above result, we infer that the Riemann-Liouville derivative of unity is not zero. We also need the Weyl fractional operator defined by where we define the Fourier transform as Following the convention initiated by Compte (1996) we suppress the imaginary unit in Fourier space by adopting a slightly modified form of the above result in our investigations (Metzler and Klafter, 2000,p.59, A.12).
We now proceed to discuss the various applications of Mittag-Leffler functions in applied sciences. In order to discuss the application of Mittag-Leffler function in kinetic equations, we derive the solution of two kinetic equations in the next section.

Application in kinetic equations
Theorem 13.1. If ℜ(ν) > 0 then the solution of the integral equation is given by where E ν (t) is the Mittag-Leffler function defined in (1.1).
Proof. Applying Laplace transform to both sides of (13.1) and using (12.12) it gives By virtue of the relation it is seen that This completes the proof of Theorem 13.1.
Remark 13.1. If we apply the operator 0 D ν t from the left to (13.1) and make use of the formula we obtain the fractional diffusion equation whose solution is also given by (13.6).
Remark 13.2. We note that Haubold and Mathai (2000) have given the solution of (13.1) in terms of the series given by (13.5). The solution in terms of the Mittag-Leffler function is given in Saxena, Mathai and Haubold (2002).
Alternate procedure. We now present an alternate method similar to that followed by Al-Saqabi and Tuan (2006) for solving some differintegral equations, also, see  for details.
Applying the operator (−c ν ) m 0 D −mν t to both sides of (13.1) we find that which can be written as Simplifying the above equation by using the result where min{ℜ(ν), ℜ(µ)} > 0, we obtain for m = 0, 1, 2, ... Rewriting the series on the right in terms of the Mittag-Leffler function, it yields the desired result (13.6). The next theorem can be proved in a similar manner.
Proof. Applying Laplace transform to both sides of (13.12) and using (1.11), it gives (13.14) Using the relation (13.4), it is seen that . (13.15) This completes the proof of Theorem 13.2.
Alternate procedure. We now give an alternate method similar to that followed by Al-Saqabi and Tuan (2006) for solving the differintegral equations. Applying the operator (−c ν ) m 0 D −mν t to both sides of (13.12), we find that for m = 0, 1, 2, .... Summing up the above expression with respect to m from 0 to ∞, it gives Simplifying by using the result (13.10) we obtain ; m = 0, 1, 2, ... (13.17) Rewriting the series on the right of (13.17) in terms of the generalized Mittag-Leffler function, it yields the desired result (13.5). Next we present a general theorem given by .
Theorem 13.3. If c > 0, ℜ(ν) > 0 then for the solution of the integral equation where f (t) is any integrable function on the finite interval [0, b], there exists the formula where H 1,1 1,2 (·) is the H-function defined by (8.1). The proof can be developed by identifying the Laplace transform of N () + c ν 0 D −ν t N (t) as an H-function and then using the convolution property for the Laplace transform. In what follows, E δ β,γ (·) will be employed to denote the generalized Mittag-Leffler function, defined by (11.1).

Note 13.1. For an alternate derivation of this theorem see Saxena and Kalla (2008).
Next we will discuss time-fractional diffusion.
14. Application to time-fractional diffusion Theorem 14.1. Consider the following time-fractional diffusion equation .

(14.2)
Proof. In order to find a closed form representation of the solution in terms of the H-function, we use the method of joint Laplace-Fourier transform, defined bỹ where, according to the convention followed, ∼ will denote the Laplace transform and * the Fourier transform.
Applying the Laplace transform with respect to time variable t, Fourier transform with respect to space variable x and using the given condition, we find that Inverting the Laplace transform, it yields where E α (·) is the Mittag-Leffler function defined by (1.1). In order to invert the Fourier transform, we will make use of the integral Fractional space-diffusion will be discussed in the next section.

Application to fractional-space diffusion
Theorem 15.1. Consider the following fractional space-diffusion equation where D is the diffusion constant, ∂ α ∂x α is the operator defined by (12.20) and N (x, t = 0) = δ(x) is the Dirac delta function and lim x→±∞ N (x, t) = 0. Then its fundamental solution is given by .

(15.2)
The proof can be developed on similar lines to that of the theorem of the preceding section.

Application to fractional reaction-diffusion model
In the same way, we can establish the following theorem, which gives the fundamental solution of the reaction-diffusion model given below.
Theorem 16.1. Consider the following reaction-diffusion model is the Dirac delta function. Then for the solution of (16.1) there holds the formula .

(16.2)
For details of the proof, the reader is referred to the original paper by .
For the solution of the fraction reaction-diffusion equation

Application to nonlinear waves
It will be shown in this section that by the application of the inverse Laplace transforms of certain algebraic functions derived in Saxena, , we can establish the following theorem for nonlinear waves.
Theorem 17.1. Consider the fractional reaction-diffusion equation where ν 2 is a diffusion constant, ζ is a constant which describes the nonlinearity in the system, and φ(x, t) is nonlinear function which belongs to the area of reaction-diffusion, then there holds the following formula for the solution of (17.1).

Generalized Mittag-Leffler type functions
The multiindex (m-tuple) Mittag-Leffler function is defined in Kiryakova (2000) by means of the power series .
The following theorem is proved by Kiryakova (2000, p.244) which shows that the multiindex Mittag-Leffler function is an entire function and also gives its asymptotic estimate, order and type.
It is interesting to note that for m = 2, (18.2) reduces to the generalized Mittag-Leffler function considered by Dzherbashyan (1960) denoted by φ ρ1,ρ2 (z; µ 1 , µ 2 ) and defined in the following form (Kiryakova, 1994, Appendix) , (18.5) and shown to be an entire function of order Another generalization of the Mittag-Leffler function is recently given by Sharma (2008) in terms of the M-series defined by Remark 18.1. According to Saxena (2009), the M-series discussed by Sharma (2008) is not a new special function. It is, in disguise, a special case of the generalized Wright function p ψ q (z), which was introduced by Wright (1935), as shown below. 1), ..., (a p , 1), (1, 1) .
Fractional integration and fractional differentiation of the M-series are discussed by Sharma (2008). The two results proved in Sharma (2008) for the function defined by (18.9) are reproduced below. For ℜ(ν) > 0 and for ℜ(ν) < 0

Mittag-Leffler Statistical Distribution and Its Properties
A statistical distribution in terms of the Mittag-Leffler function E α (y) was defined by Pillai (1990) in terms of the distribution function or cumulative density function as follows: and G y (y) = 0 for y ≤ 0. Differentiating on both sides with respect to y we obtain the density function f (y) as follows: by replacing k by k + 1 where E α,β (x) is the generalized Mittag-Leffler function.
It is straightforward to observe that for the density in (19.1.2) the distribution function is that in (19.1.1). The Laplace transform of the density in (19.1.2) is the following: Note that (19.1.3) is a special case of the general class of Laplace transforms discussed in Section 2.3.7 ]. From (19.1.3) one can also note that there is a structural representation in terms of positive Lévy distribution. A positive Lévy random variable u > 0, with parameter α is such that the Laplace transform of the density of u > 0 is given by e −t α . That is, where E(·) denotes the expected value of (·) or the statistical expectation of (·). When α = 1 the random variable is degenerate with the density function Consider an exponential random variable with density function and with the Laplace transform L f1 (t). Proof. For establishing this result we will make use of a basic result on conditional expectations, which will be stated as a lemma.  Now, let x be an arbitrary positive random variable having Laplace transform, denoted by L x (t) where L x (t) = ψ(t). Then from (19.1.9) we have For example, if y is a random variable whose density has the Laplace transform, denoted by L y (t) = φ(t), with φ(tx 1 α ) = xφ(t), and if x is a real random variable having the gamma density, and f x (x) = 0 elsewhere, and if x and y are statistically independently distributed and if u = yx 1 α then the Laplace transform of the density of u, denoted by L u (t) is given by which will be Γ(β − t)/Γ(β) for the density in (19.1.13). Hence where ψ(·) is the psi function of (·), see Mathai (1993) for details. Hence by taking the limits t → 0 where γ is Euler's constant, see Mathai (1993) for details.

Mellin-Barnes representation of the Mittag-Leffler density
Consider the density function in (19.1.2). After writing in series form and then looking at the corresponding Mellin-Barnes representation we have the following: [by expanding as the sum of residues at the poles of Γ(1 − 1 α + s α )] Here the point s = 0 is removable. By taking the Laplace transform of g(x) from (19.2.1) we have

Generalized Mittag-Leffler density
Consider the generalized Mittag-Leffler function Laplace transform of g 1 (x) is the following: In fact, this is a special case of the general class of Laplace transforms connected with Mittag-Leffler function considered in . (19.2.6) can be written as a Mellin-Barnes integral and then as an H-function.

Mittag-Leffler Density as an H-function
Since g and g 1 are represented as inverse Mellin transforms, in the Mellin-Barnes representation, one can obtain the (s − 1)-th moments of g and g 1 from (19.3.2). That is, for 1 − α < ℜ(s) < 1, 0 < α ≤ 1, obtained by putting η = 1 in (19.3.4) also. Since Its inverse Mellin transform is then which is the one-parameter gamma density and for η = 1 it reduces to the exponential density. Hence the generalized Mittag-Leffler density g 1 can be taken as an extension of a gamma density such as the one in (19.3.7) and the Mittag-Leffler density g as an extension of the exponential density for η = 1. Is there a structural representation for the random variable giving rise to the Laplace transform in (19.2.4) corresponding to (19.1.10)? The answer is in the affirmative and it is illustrated in (19.1.14).
Hence g(x) is a density function for a positive random variable x. Note that from the series form for the Mittag-Leffler function it is not possible to show that ∞ 0 g(x)dx = 1 directly.

Structural Representation of the Generalized Mittag-Leffler Variable
Let u be the random variable corresponding to the Laplace transform (19.2.7) with t α replaced by δt α and γ by η. Let u be a positive Lévy variable with the Laplace transform e −t α , 0 < α ≤ 1 and let v be a gamma random variable with parameters η and δ or with he Laplace transform (1 + δt) −η , η > 0, δ > 0. Let u and v be statistically independently distributed. where w is a generalized Mittag-Leffler variable with Laplace transform (1+δt α ) −β , |δt α | < 1, where ∼ means 'distributed as' or both sides have the same distribution.
Proof. Denoting the Laplace transform of the density of w by L w (t) and treating it as an expected value for ℜ(s) > 1 − αη, 0 < α ≤ 1, η > 0. Comparing (19.4.3) and (19.4.4) we have the (s − 1)-th moment of a Lévy variable . (19.4.6) Hence for ρ = s − 1 we have This is (19.4.5) and hence both the representations are one and the same.

A Pathway from Mittag-Leffler Distribution to Positive Lévy Distribution
Consider the function .
Thus x = a 1 α y where y is a generalized Mittag-Leffler variable. The Laplace transform of f is given by the following: If η is replaced by η q−1 and a by a(q − 1) with q > 1 then we have a Laplace transform which is the Laplace transform of a constant multiple of a positive Lévy variable with parameter α. Thus q here creates a pathway of going from the general Mittag-Leffler density f to a positive Lévy density f 1 with parameter α, the multiplying constant being (aη) 1 α . For a discussion of a general rectangular matrix-variate pathway model see Mathai (2005). The result in (19.5.3) can be put in a more general setting. Consider an arbitrary real random variable y with the Laplace transform, denoted by L y (t), and given by L y (t) = e −φ(t) (19.5.4) where φ(t) is a function such that φ(tx γ ) = xφ(t), φ(t) ≥ 0, lim t→0 φ(t) = 0 for some real positive γ. Let u = yx γ (19.5.5) where x and y are independently distributed with y having the Laplace transform in (19.5.4) and x having a two-parameter gamma density with shape parameter β and scale parameter δ or with the Laplace transform L x (t) = (1 + δt) −β . If δ is replaced by δ(q − 1) and β by β q−1 with q > 1 then we get a path through q. That is, when q → 1 + , L u (t) = [1 + δ(q − 1)φ(t)] − β q−1 → e −δβφ(t) = e −φ(δ γ β γ t) . (19.5.8) If φ(t) = t α , 0 < α ≤ 1 then L u (t) = e −(δβ) γα t α which means that u goes to a constant multiple of a positive Lévy variable with parameter α, the constant being (δβ) γ .

Linnik or α-Laplace Distribution
A Linnik random variable is defined as that real scalar random variable whose characteristic function is given by φ(t) = 1 1 + |t| α , 0 < α ≤ 2, −∞ < t < ∞. (19.6.1) For α = 2, (19.6.1) corresponds to the characteristic function of a Laplace random variable and hence Pillai (1995) called the distribution corresponding to (19.6.1) as the α-Laplace distribution. For positive variable, (19.6.1) reduces to the characteristic function of a Mittag-Leffler variable. Infinite divisibility, characterizations, other properties and related materials may be seen from the review paper Jayakumar and Suresh (2003) and the many references therein, Pakes (1998) and Mainardi and Pagnini (2008). Multivariate generalization of Mittag-Leffler and Linnik distributions may be seen from Lim and Teo (2009). Since the steps for deriving results on Linnik distribution are parallel to those of the Mittag-Leffler variable, further discussion of Linnik distribution is omitted.

Multivariable Generalization of Mittag-Leffler, Linnik and Lévy Distributions
A multivariate Linnik distribution can be defined in terms of a multivariate Lévy vector. Let T ′ = (t 1 , ..., t p ), X ′ = (x 1 , ..., x p ), prime denoting the transpose. A vector variable having positive Lévy distribution is given by the characteristic function where the p × 1 vector X, having a multivariable Lévy distribution with parameter α, and y a real scalar gamma random variable with the parameters δ and β, are independently distributed. Then the characteristic function of the random vector variable u is given by the following:

Linear First Order Autoregressive processes
Consider the stochastic process x n = e n , with probability p, 0 ≤ p ≤ 1 e n + ax n−1 with probability 1 − p, 0 < a ≤ 1. which defines class L distributions, for all a, 0 < a < 1. When p = 0, (20.2.3) implies that the innovation sequence {e n } belongs to class L distributions. Then (20.2.3) can lead to two autoregressive situations, the first order exponential autoregressive process EAR(1) and the first order Mittag-Leffler autoregressive process M LAR(1).

Concluding remarks
The various Mittag-Leffler functions discussed in this paper will be useful for investigators in various disciplines of applied sciences and engineering. The importance of Mittag-Leffler function in physics is steadily increasing. It is simply said that deviations of physical phenomena from exponential behavior could be governed by physical laws through Mittag-Leffler functions (power-law). Currently more and more such phenomena are discovered and studied.
It is particularly important for the disciplines of stochastic systems, dynamical systems theory, and disordered systems. Eventually, it is believed that all these new research results will lead to the discovery of truly nonequilibrium statistical mechanics. This is statistical mechanics beyond Boltzmann and Gibbs. This nonequilibrium statistical mechanics will focus on entropy production, reaction, diffusion, reaction-diffusion, etc and may be governed by fractional calculus.
Right now, fractional calculus and H-function (Mittag-Leffler function) are very important in research in physics.