Marichev-Saigo-Maeda fractional calculus operators, Srivastava polynomials and generalized Mittag-Leffler function

The aim of this paper is to evaluate four theorems for generalized fractional integral and derivative operators, applied on the product of Srivastava polynomials and generalized Mittag-Leffler function. The results are expressed in terms of generalized Wright function. Further, we also point out their relevance with the known results. Subjects: Science; Mathematics & Statistics function; Technology; Engineering & Technology


PUBLIC INTEREST STATEMENT
The Mittag-Leffler functions are very useful almost in all areas of applied Mathematics, that provides solutions to a number of problems formulated in terms of fractional order differential, integral and difference equations; therefore, it has recently become a subject of interest for many authors in the field of fractional calculus and its applications. In this paper, we have evaluated four theorems for generalized fractional integral and derivative operators, applied on the product of Srivastava polynomials and generalized Mittag-Leffler function and also point out their relevance with the known results.

Introduction
The Mittag-Leffler functions are important special functions, that provides solutions to number of problems formulated in terms of fractional order differential, integral and difference equations; therefore, it has recently become a subject of interest for many authors in the field of fractional calculus and its applications. For detailed account of fractional calculus operators along with their properties and applications, one may refer to the research monographs by Kilbas, Srivastava, and Trujillo (2006), Kiryakova (1994), Miller and Ross (1993), Srivastava and Saigo (1987), Srivastava and Saxena (2001) and recent papers , Mishra, Agarwal, and Sen (2016), , Mishra, Srivastava, and Sen (2016), Purohit (2013) and .
The Swedish mathematician Mittag-Leffler (1903) introduced the function E (z), defined by: A further, two-index generalization of this function was studied by Wiman (1905) as: where ℜ( ) > 0 and ℜ( ) > 0. Prabhakar (1971) introduced the generalization of Mittag-Leffler function E , (z) in the form where , , ∈ ℂ, ℜ( ) > 0. Further, it is an entire function of order Re( ) −1 (see Prabhakar, 1971, p. 7). Shukla and Prajapati (2007) (see also Srivastava & Tomovski, 2009) denotes the generalized Pochhammer symbol, which in particular reduces to It is remarked that certain much more general functions of the Mittag-Leffler type have already been investigated in the literature rather systematically and extensively, but for the purpose of this paper we use the function given by (4) only.
The generalized Wright function p q (z) defined for z ∈ ℂ a i , b j ∈ ℂ, and A i , B j ∈ ℜ(A i , B j ≠ 0; i = 1, 2, … , p; j = 1, 2, … , q) is given by the series where Γ(z) is the Euler gamma function and the function (5) was introduced by Wright (1935) and is known as generalized Wright function, for all values of the argument z, under the condition: (1) For detailed study of various properties, generalization and application of Wright function and generalized Wright function, we refer to paper (for instance, see Wright, 1935. The Srivastava polynomials defined by Srivastava (1968, p. 1, Equation (1)) in the following manner: where u is an arbitrary positive integer and the coefficients A w.s (w, s) ≥ 0 are arbitrary constants, real or complex.
On account of success of the Saigo operators (Saigo, 1978(Saigo, , 1979, in their study on various function spaces and their application in the integral equation and differential equations, Saigo and Maeda (1998) introduced the following generalized fractional and differential operators of any complex order with Appell function F 3 (⋅) in the kernel, as follows: Let , � , , � , ∈ ℂ and x > 0, then the generalized fractional calculus operators (the Marichev-Saigo-Maeda operators) involving the Appell function, or Horn's F 3 -function are defined by the following equations: For the definition of the Appell function F 3 (⋅) the interested reader may refer to the monograph by Srivastava and Karlsson (1985) (see Erdélyi, Magnus, Oberhettinger, and Tricomi (1953), Prudnikov, Brychkov, and Marichev (1992) and Samko, Kilbas, and Marichev (1993)).
Following Saigo and Maeda (1998), the image formulas for a power function, under operators (8) and (10), are given by: Here, we used the symbol Γ ⋯ ⋯ representing the fraction of many Gamma functions.

Left-sided generalized fractional integration of product of polynomial and generalized Mittag-Leffler function
In this section, we establish image formulas for the product of Srivastava polynomial and general- Proof On using (4) and (7), writing the function in the series form, the left-hand side of (16), leads to Now, upon using the image formula (14), which is valid under the conditions stated with Theorem 2.1, we get Interpreting the right-hand side of the above equation, in view of the definition (5), we arrive at the result (16).

Right-sided generalized fractional integration of product of polynomial and generalized Mittag-Leffler function
In this part, we establish image formulas for the product of Srivastava polynomial and generalized Mittag-Leffler function involving right-sided operators of Marichev-Saigo-Meada fractional integral operators (10), in term of the generalized Wright function. These formulas are given by the following theorems: Proof On using (4) and (7), the left-hand side of (23), can be written as: which on using the image formula (15), arrive at Interpreting the right-hand side of the above equation, in view of the definition (5), we arrive at the result (23).

Left-sided generalized fractional differentiation of product of polynomial and generalized Mittag-Leffler function
Now, we shall establish image formulas for the product of Srivastava polynomial and generalized Mittag-Leffler function involving left-sided operators of Marichev-Saigo-Meada fractional differentiation operators (12) in term of the generalized Wright function. These formulas are given by the following theorems: Proof On using (4) and (7), writing the function in the series form, the left-hand side of (29), leads to Now, upon using the image formula (14), which is valid under the conditions stated with Theorem 2.3, we get