Research Article | Open Access
Fractional Integral and Derivative Formulas by Using Marichev-Saigo-Maeda Operators Involving the S-Function
We establish fractional integral and derivative formulas by using Marichev-Saigo-Maeda operators involving the S-function. The results are expressed in terms of the generalized Gauss hypergeometric functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals and derivatives are presented. Also we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance.
1. Introduction and Preliminaries
In recent times, the fractional calculus is the most fast growing subject of mathematical analysis. It is concern with applied mathematics that deals with integrals and derivatives of arbitrary orders. The fractional calculus operator linking diverse special functions has found substantial significance and applications in a variety of subfields of applicable mathematical analysis. Numerous applications of fractional calculus can be found in astrophysics, turbulence, nonlinear biological systems, fluid dynamics, stochastic dynamical system, plasma physics and nonlinear control theory, image processing, and quantum mechanics. Since last four decades, a number of workers like [1–9] so forth have studied, in depth, the properties, applications, and diverse extensions of a range of operators of fractional calculus. A comprehensive account of generalized fractional calculus operators along with their properties and applications can be found in [10–13] and also the research monographs [14, 15] and so forth.
On relation of success of the Saigo operators [16, 17], in their study on various function spaces and their application in the integral equation and differential equations, Saigo and Maeda  introduced the following generalized fractional integral and differential operators of any complex order with Appell function (.) in the kernel which is extension of Marichev , as follows.
Let and 0, then the generalized fractional calculus formulas (the Marichev-Saigo-Maeda operators) involving the Appell function or Horn’s -function are defined by the following equations:andIn (1) and (3), denotes Appell function  in two variables defined as
Remark. The Appell function defined in above equation reduces to Gauss hypergeometric function as given in the following relations:andandIn view of the above reduction formula as given in (10), the general fractional calculus operators reduce to the Saigo operators  defined as follows:andwhere , a special case of the generalized hypergeometric function, is the Gauss hypergeometric function and the function is so constrained that the integrals in (13) and (15) converge.
If we take in (13), (15), (17), and (18), we obtain the Erdélyi-Kober fractional integral and derivative operators [11, 21], defined as follows:andWhen , then operators in (13), (15), (17), and (18) give the Riemann-Liouville and the Weyl fractional integral and derivative operators [11, 22] are defined as follows:and
Lemma 1. Let , and be such that , then the following formulas hold true:and
Lemma 2. Let , and be such that , then the following formulas hold true:and
Lemma 3. Let , and be such that , then the following formulas hold true:and
Lemma 4. Let , and be such that , then the following formulas hold true:and
The S-function is defined by Saxena and Daiya  as, , and The -Pochhammer symbol and -gamma function introduced by Diaz and Pariguan  are as follows:and the relation with the classical Euler’s gamma function is as follows:where , and . For further details of -Pochhammer symbol and -special functions one can refer to the papers by Romero et al. .
Special Cases(1)When in (43), the S-function reduced to generalized -Mittag-Leffler function, defined by Saxena et al. :(2)When the S-function reduced to generalized K-function, defined by Sharma :(3)When , the S-function reduced to generalized M-series defined by Sharma and Jain :Here, our aim is to establish composition formula of Marichev-Saigo-Maeda fractional integral and derivative operators of the product of S-function. The main formulas obtained here are represented in terms of the generalized Wright function defined for , , and ; which is given by the serieswhere is the Euler gamma function and the function was introduced by Wright  and is known as generalized Wright function. Several theorems are on the asymptotic expansion of for all values of the argument , under the conditionFor detailed study of various properties, generalization and application of Wright function, and generalized Wright function, we refer to, for instance, [30, 31].
3. Approach to Fractional Calculus
Throughout this paper, we assume that , , , and , such that , , and Further, let the constants satisfy the condition , and ; , such that condition (53) is also satisfied.
3.1. Left- and Right-Sided Generalized Fractional Integration of S-Function
In this section, we establish image formulas for the product of S-function involving left- and right-sided operators of Marichev-Saigo-Meada fractional integral operators (1) and (3), respectively, in terms of the generalized Wright function. These formulas are given by the following theorems.
Theorem 5. Let , , then the generalized fractional integration of the product of S-function is given by
Proof. On using (43), writing the function in the series form, the left-hand side of (55) leads toBy interchanging the order of integration and summation, we reduce the right side of (57) toBy applying Lemma 1 (see (27)) in (58), we getBy applying (44) and (45) in (60) we getInterpreting the right-hand side of the above equation, in view of definition (52), we arrive at result (55).
Theorem 6. Let , , then the generalized fractional integration of the product of S-function is given by
3.1.1. Special Cases
Corollary 7. Let , , then the generalized fractional integration of the product of S-function is given by
Corollary 8. Let , , then the generalized fractional integration of the product of S-function is given by
Corollary 9. Let , , then the generalized fractional integration of the product of S-function is given by
Corollary 10. Let , , then the generalized fractional integration of the product of S-function is given by
Corollary 11. Let , , then the generalized fractional integration of the product of S-function is given by
Corollary 13. Let , , then the generalized fractional integration of the S-function is given by