Research Article | Open Access
Fractional Calculus of the Generalized Mittag-Leffler Type Function
We introduce and study a new function called -function, which is an extension of the generalized Mittag-Leffler function. We derive the relations that exist between the -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.
1. 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.
In 1903, the Swedish mathematician Mittag-Leffler [1, 2] introduced the function , defined by The Mittag-Leffler function was studied by Wiman  who defined the function as follows: The function is now known as Wiman function, which was later studied by Agarwal  and others.
The generalization of (2) was introduced by Prabhakar  in terms of the series representation where is Pochhammer’s symbol, defined by Shukla and Prajapati  defined and investigated the function as where and denotes the generalized Pochhammer symbol which in particular reduces to Srivastava and Tomovski  introduced and investigated a further generalization of (3), which is defined in the following way: which, in the special case when and , is given by (5).
Remark 1. A detailed account of Mittag-Leffler functions and their applications can be found in the monograph by Haubold et al. .
1.1. Fractional Integrals and Derivatives
An interesting and useful generalization of both the Riemann-Liouville and Erdélyi-Kober fractional integration operators is introduced by Saigo  in terms of Gauss's hypergeometric function as follows: where , , and , and the generalized fractional derivative of a function where .
2. The -Function
The -function is introduced by the authors as follows: where , ; ; , and are the Pochhammer symbols. The series (12) is defined when none of the parameters ’s, , 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 . It can also converge in some cases if we have and . Let ; it can be shown that if and the series is absolutely convergent for , in order convergent for when and divergent for when .
2.1. Special Cases of the -Function
(vii) If we set in (12), then the -function can be represented in the Wright generalized hypergeometric function  and the -function [13, 14] as given below: where -function is as defined in the monograph by Mathai and Saxena .
(viii) If we set and in (12), then we obtain another special case of -function in terms of the Wright generalized hypergeometric function as given below:
3. Relations with Generalized Fractional Calculus Operators
In this section we derive two theorems relating to generalized fractional integrals and derivative of the -function.
Theorem 2. Let , , , and ; then there holds the relation
Proof. Following the definition of Saigo fractional integral  as given in (10), we have the following relation:
by virtue of (12), we obtain
Interchanging the order of integration and evaluating the inner integral with the help of Beta function, we arrive at
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 .
This completes the proof of Theorem 2.
If we put , then we obtain following Corollary concerning Riemann-Liouville fractional integral operator .
Corollary 3. Let , , , and , then there holds the relation
Theorem 4. Let , , , and , then there holds the relation
Proof. Following the definition of Saigo fractional derivative as given in (11), we have the following relation:
By virtue of (12), we obtain Interchanging the order of integration and evaluating the inner integral with the help of Beta function, we arrive at Here , and by using , where in the above expression, we obtain the right-hand side of (27). This shows that a Saigo fractional derivative of the -function is again the -function with increased order .
This completes the proof of Theorem 4.
If we put , then we obtain following Corollary concerning Riemann-Liouville fractional derivative operator .
Corollary 5. Let , , , and ; then there holds the relation
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 . 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors wish to thank the referees for their useful suggestions for the improvement of the paper. The authors are thankful to Professor R. K. Saxena for giving useful suggestions, which led to the present form of the paper.
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Copyright © 2014 Dinesh Kumar and Sunil Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.