Fractional Calculus of the Generalized Mittag-Leffler Type Function

We introduce and study a new function called R-function, which is an extension of the generalized Mittag-Leffler function. We derive the relations that exist between the R-function and Saigo fractional calculus operators. Some results derived by Samko et al. (1993), Kilbas (2005), Kilbas and Saigo (1995), and Sharma and Jain (2009) are special cases of the main results derived in this paper.


Introduction and Preliminaries
The Mittag-Leffler function has gained importance and popularity during the last one and a half decades due to its direct involvement in the problems of physics, biology, engineering, and applied sciences. Mittag-Leffler function naturally occurs as the solution of fractional order differential equations and fractional order integral equations.
The function , ( ) is now known as Wiman function, which was later studied by Agarwal [4] and others.
The generalization of (2) was introduced by Prabhakar [5] in terms of the series representation where ( ) is Pochhammer's symbol, defined by Shukla and Prajapati [6] defined and investigated the function Srivastava and Tomovski [7] introduced and investigated a further generalization of (3), which is defined in the following way: , which, in the special case when = ( ∈ (0, 1) ∪ ) and min{Re( ), Re( )} > 0, is given by (5).
It is an entire function of order = [Re( )] −1 . Some special cases of (3) are here 1 1 denotes an hypergeometric function; see also 2 1 in (10).

Remark 1.
A detailed account of Mittag-Leffler functions and their applications can be found in the monograph by Haubold et al. [8].

The -Function
The -function is introduced by the authors as follows: where , , ∈ , Re( ) > max{0, Re( ) − 1}; Re( ) > 0; ( ) , and ( ) are the Pochhammer symbols. The series (12) is defined when none of the parameters 's, = 1, , is a negative integer or zero. If any parameter is a negative integer or zero, then the series (12) terminates to a polynomial in, and the series is convergent for all if < +1. It can also converge in some cases if we have = + 1 and | | = 1. Let = ∑ =1 − ∑ =1 ; it can be shown that if Re( ) > 0 and = + 1 the series is absolutely convergent for | | = 1, in order convergent for = −1 when 0 ≤ Re( ) < 1 and divergent for | | = 1 when 1 ≤ Re( ).

Special Cases of the -Function. (i) If we set
where , , ( ) is the generalized Mittag-Leffler function introduced by Srivastava and Tomovski [7]; compare (5).
(vii) If we set = = 1 in (12), then the -function can be represented in the Wright generalized hypergeometric function [12] ( ) and the -function [13,14] as given below: where -function is as defined in the monograph by Mathai and Saxena [14].
(viii) If we set = = 0 and = = 1 in (12), then we obtain another special case of -function in terms of the Wright generalized hypergeometric function as given below: (ix) If we set = = = = 1 in (12), then the -function reduces to the generalized hypergeometric function (see for detail [11,15,16]) as given below:

Relations with Generalized Fractional Calculus Operators
In this section we derive two theorems relating to generalized fractional integrals and derivative of the -function.

(22)
Proof. Following the definition of Saigo fractional integral [17] as given in (10), we have the following relation: The interchange of the order of summation is permissible under the conditions stated along with the theorem. This shows that a Saigo fractional integral of the -function is again the -function with increased order ( + 2, + 2). This completes the proof of Theorem 2.
If we put = − , then we obtain following Corollary concerning Riemann-Liouville fractional integral operator [16].
This completes the proof of Theorem 4.
If we put = − , then we obtain following Corollary concerning Riemann-Liouville fractional derivative operator [16]. (31) Remark 6. A number of known and new results can be obtained as special cases of Theorems 2 and 4, but we do not mention them here on account of lack of space.

Conclusion
In this paper we derive a new generalization of Mittag-Leffler function and obtain the relations between the -function and Saigo fractional calculus operators. The results are also extension of work done by Sharma [18]. The provided results are new and have uniqueness identity in the literature. A number of known and new results are special cases of our main findings.