Fractional integral operator associated with extended Mittag-Leffler function

Popular literature of Special Functions includes the generalization and extensions of functions like gamma function, beta function, Mittag-Leffler function, hypergeometric function and confluent hypergeometric function etc. This sequel deals with the extension of Mittag-Leffler functions and its properties. We aim to find the composition of fractional integration formula known as Ҏ 𝛿 − transform with the extended Mittag-Leffler function. Some special cases and corollaries are pointed out which follow from our main results .

For the present sequel, we consider the following definitions and the related work.
be two power series then the Hadamard Product of the two series is defined as (Pohlen, 2009): Where and ℎ are the radii of convergence of two series ( ) and ℎ( ) respectively. Therefore, in general, ≥ • ℎ . It is to be noted that if one of the power series is an analytical function, then the Hadamard product series is also an analytical function.

Definition:
The Ҏ -transform Ҏ [ ( ); ] of a function ( ) where is a complex variable defined by (Kumer, 2013): provided that the sufficient existence conditions given by Lemma below be satisfied.
The power function of the transform is given below.
It is to be noted that when transform is converted into classical Laplace transform. The integral form of a classical Laplace transform is given below, for reference see Sneddon (1979).
where , 1 and 2 are constant. must be finite and 1 , 2 may or may not be finite. Elzaki Transform for a function ( ) is defined as where the variable is used to factorize the variable in the argument of the function .

Ҏ -transform of extended Mittag-Leffler function
In this section, we consider the composition of Ҏ -integral transform of pathway type with where 0 1 is a well known Gauss hypergeometric function (Rainville, 1971).
Proof: For the sake of convenience, let us denote the left hand side by Ω, we have Due to uniform convergence, order of integration and summation can be changed and then replacing the variable by + in (12) Thus, the last expression can easily be obtained by means of Hadamard Product as in (9). If we select the sequence ҝ = 1, ( 0 ) then the proposed function reduces to the definition of Özarslan and Yilmaz (2014).
If we set the value of the parameters, = = 1the extension of the proposed function takes the form of extended confluent hypergeometric function then the main theorem reduces to the following corollary.
Corollary 2: Let , , ∁ then the following relation holds true. Further, for setting = 0then function will become Prabhakar's definition and we get the results as in Kumar (2013). Setting = = 1and the bounded sequences ҝ = 1then the expression yields the result for extended confluent hypergeometric function defined by Chaudhry et al. (2004). Further, for setting = 0 and = = 1then the result is for the classical confluent hypergeometric function.

Further special cases and concluding remarks
It is to be noted that Ҏ -transform reduces to classical Laplace transform by converting the variable then we get integral involving Laplace transform stated in corollaries.
Hence, it is easy to see that two integrals have the following relationship.

Conclusion
It is noted that the extended Mittag-Leffler function defined by Parmar (2015) is more general in nature and various generalized types of Mittag-Leffler function and confluent hypergeometric functions defined in literature can easily be derived through the extended form. Similarly, the Ҏ −transform defined by Kumar (2013) (fractional integral operator) enable us to convert the table of Laplace transform and the Elzaki transform into the corresponding transform and vice versa.