Consider the function defined for $\alpha _{1},\ \alpha _{2} \in \mathbb{R}$ (α 1 2 +α 2 2 ≠ 0) and $\beta _{1},\beta _{2} \in \mathbb{C}$ by the series 

$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$

(6.1.1)

Such a function with positive α 1 > 0, α 2 > 0 and real $\beta _{1},\beta _{2} \in \mathbb{R}$ was introduced by Dzherbashian [Dzh60].

Multi-index Mittag-Leffler Functions

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.

https://downloads.hindawi.com/journals/jam/2011/298628.pdf

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 paper is devoted to the study of the function E γ ρ,μ(z) defined for complex ρ, μ, γ (Re(ρ) > 0) by which is a generalization of the classical Mittag-Leffler function E ρ,μ(z) and the Kummer confluent hypergeometric function Φ(γ, μ; z). The properties of E γ ρ,μ(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel is studied in the space L(a, b). Compositions of the Riemann–Liouville fractional integration and differentiation operators with E γ ρ,μ,ω;a+ are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E γ ρ,μ,ω;a+. Since, for γ = 0, E 0 ρ,μ,ω;a+ coincides with the Riemann–Liouville fractional integral of order μ, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mitta...

Generalized mittag-leer function and generalized fractional calculus operators

Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel

is an entire function of order $({\rm Re}\alpha)^{-1}$ and contains several well-known special functions as particular cases. We define a linear operator $\mathfrak{C}(\alpha, \beta;\rho;\lambda)$ on a space $L$ of functions by the integral in (1.2) and employ an operator of fractional integration $I^{\mu}$ : $L\rightarrow L$ to prove results on $\mathfrak{C}(\alpha, \beta;\rho;\lambda)$ ; these results are subsequently used to discuss theorems on the solutions of (1.2). The technique used can be apPlied to obtain analogous results on the integral equation

Tilak Raj Prabhakar

Papers

A singular integral equation with a generalized mittag leffler function in the kernel