A Class of Extended Mittag – Leffler Functions and Their Properties Related to Integral Transforms and Fractional Calculus

In a joint paper with Srivastava and Chopra, we introduced far-reaching generalizations of the extended Gammafunction, extended Beta function and the extended Gauss hypergeometric function. In this present paper, we extend the generalized Mittag–Leffler function by means of the extended Beta function. We then systematically investigate several properties of the extended Mittag–Leffler function, including, for example, certain basic properties, Laplace transform, Mellin transform and Euler-Beta transform. Further, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag–Leffler function are investigated. Some interesting special cases of our main results are also pointed out.

In 1971, Prabhakar [18] introduced the three-parameter generalization of Equation (14) as: called usually the Prabhakar function.Further, various authors studied and investigated generalized Mittag-Leffler functions (see, for details, [19][20][21][22][23][24][25]).Motivated essentially by the demonstrated potential for applications of these extended hypergeometric functions, we extend the generalized Mittag-Leffler function (Equation ( 15)) by means of the extended Beta function B ({κ } ∈N 0 ) (x, y ; p) defined by Equation (3) and investigate certain basic properties, including differentiation formulas and the integral property, Laplace transform, Euler-Beta transform and Mellin transform with their several special cases and relationships with generalized hypergeometric function p F q and H-function.Further, certain relations between the extended generalized Mittag-Leffler function and the Riemann-Liouville fractional integrals and derivatives are investigated.Some interesting special cases of our main results are also considered.
Remark 2. The special case for α = β = 1 in Equations ( 16)-( 18) can be expressed in terms of the extended confluent hypergeometric functions as: and: In this section, we obtain certain basic properties, including the differentiation formula and the integral property of the extended generalized Mittag-Leffler function in Equation ( 16).Theorem 1.The following differentiation formula for the extended generalized Mittag-Leffler function in Equation ( 16) holds true: In particular, we have: Proof.Using the definition (Equation ( 16)) in right-hand side of Equation ( 19), we have: The relation Equation ( 20) follows from Equation ( 19) when p = 0 or for κ = 0 ( ∈ N).
Theorem 2. The following derivative formulas for the extended generalized Mittag-Leffler function in Equation ( 16) are satisfied: In particular, we have: Proof.Using Equation ( 16) and employing term-wise differentiation m times on the left-hand side of Equation ( 21) under the summation sign, which is possible in accordance with the uniform convergence of the series in Equation ( 16), we get: The special cases of Equation ( 21) when p = 0 or for κ = 0 ( ∈ N) are easily seen to yield Equation ( 22).Corollary 1.The following integral property for the extended generalized Mittag-Leffler function in Equation ( 16) holds true: where α, β, γ, ω ∈ C; (α) > 0, (β) > 0; (p) > 0.

Mellin Transform
The Mellin transform [27] of a suitably-integrable function f (t) with index s is defined, as usual, by: whenever the improper integral in Equation ( 27) exists.
Then, for x > a, the relation holds: and Proof.By virtue of the formulas (Equations ( 41) and ( 16)), the term-by-term fractional integration and the application of the relation [34]: Next, by Equations ( 42) and ( 16), we find that: Remark 4. The special cases of the results presented here when p = 0 or for κ = 0 ( ∈ N) would reduce to the corresponding well-known results for the generalized Mittag-Leffler function (see, for details, [18] and [23]).