Marichev-Saigo Integral Operators Involving the Product of K-Function and Multivariable Polynomials

The aim of this paper to establish the Marichev-Saigo-Maeda fractional integration formula to the product of the K-function with the general class of multivariable polynomials. The results are presented in terms of the Wright generalized hypergeometric function. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Further, we point out also their relevance.


Introduction
Fractional calculus is a branch of classical mathematics, formulated in 1695.After long time, fractional calculus was not very popular among science and engineering community and was regarded as a pure mathematical realm without real applications.Now the field of fractional calculus is undergoing rapid developments with more and more convincing applications in the real world.Various authors Baleanu et al. [2], Kilbas [5], Kiryakova ([6], [7]), Kumar et al. [8], Samko et al. [14] and Suthar et al. [22] etc. investigated on the field of fractional calculus and its applications.Several motivating results which are significant to the present work are also obtained.A generalization of the hypergeometric fractional integrals for α, α ′ , β , β ′ , δ ∈ C and ℜ(δ ) ∈ C, is introduced by Marichev [9] as follows: In (1.1) and (1.2), F 3 (.)denotes the Appell function (also known as Horn function) which is introduced by Srivastava and Karlson [21] p F q (α, α ′ , β , β ′ ; δ ; These operators were generalization of Saigo fractional integral operators [12].In addition, Saigo and Maeda [13] have been studied on their properties.Considering this, the left sided and right sided generalized integration of the type (1.1) and (1.2) for a power function are as follows; ( where where An interesting further generalization of the generalized hyper-geometric function is due to Wright [23] in a series representation of the form We recall, The K-function defined by [17] as: where µ, ξ , γ ∈ , ℜ (µ) > 0, (a j ) n and (b j ) n are the Pochhammer symbols.
If any numerator parameter a jn is a negative integer or zero, then the series terminates to a polynomial in x.The series (1.8) is defined when none of parameters b jn , j = 1, 2, ..., q is a negative integer or zero.From the ratio test it is evident that the series is convergent for all x if p > q + 1.When p = q + 1 and |x| = 1, the series can converge in some cases.The corresponding Wright generalized hypergeometric function (1.6) of the generalized K-function is given by: [ (a 1 , 1), ..., (a p , 1), (γ, 1); (b 1 , 1), ..., (b q , 1), (ξ , µ); z ] . (1.9) As per Srivastava and Garg([20], pp.686, eq.( 14)), the definition of multivariable generalization of the polynomial S m n (x) is ; In which, h 1 , ..., h s ∈ Z + where as the coefficients A (L; k 1 , ..., k s ) , (L, h i ∈ N 0 , i = 1, ..., s) are arbitrarily.Chosen constants real or complex, As Srivastava [19] defined; by s = 1 on the above polynomial we obtain a polynomial of the form S m n (x).

Main Results
Here, we establish two image formulas for the K-function involving left and right sided operators of Marichev-Saigo-Maeda fractional integral operators defined in (1.1) and (1.2), results are in term of the generalized Wright function.These formulas are given by the following theorems of the product of K-function µ,ξ ,γ p K q (.) and multivariable polynomial S k 1 ,...,k s L (.) exists, under the condition then there hold the following formula: (

.11)
Proof.Let Θ be the left-hand side of (2.11), using (1.8) and (1.10) then changing the order of integration and summation, we obtain Applying (1.4) on the above equation (2.12), it becomes Interpreting the right-hand side of the above equation, in view of the definition (1.9), we arrive at the result (2.11).

.14)
The conditions of validity of the above result can be followed.

.15)
Proof.Let Θ be the left-hand side of (2.15), using (1.8) and (1.10) and then changing the order of integration and summation, we obtain Applying (1.5) on the above equation (2.16), it becomes (2.17) Interpreting the right-hand side of the above equation, in view of the definition (1.9), we arrive at the result (2.15).p K q (.) exists, and the following integral holds true: (

Conclusion
The K-function, expressed in this paper, is relatively basic in nature.Therefore, on some suitable adjustment of the parameters on function, we may obtain other special functions such as M-series, Mittag-Leffler function, Bessel -Maitland function (see, e.g., ( [1], [3], [10], [11], [15], [16] ) as its special cases,and therefore, various unified fractional integral presentations can be obtained as special cases of our results.Moreover, the results obtained in this paper also corresponds to Saigo hypergeometric fractional integrals, Weyl and Riemann-Liouville fractional integral operators as special cases.