On some Multidimensional Fractional Integral Operators Involving Multivariable I-Function

In the present paper, we study certain multidimensional fractional integral operators involving a general I-function in their kernel. We give five basic properties of these operators, and then establish two theorems and two corollaries, which are believed to be new. These basic theorems exhibit structural relationships between the multidimensional integral transforms. The oneand two-dimensional analogues of these results, which are new and of interest in themselves, can easily be deduced. Special cases of these latter theorems will give rise to certain known results obtained from time to time by several earlier authors. Introduction Fractional integral operators have been defined and studied by various authors notably by Riemann and Liouville [4], Weyl [4], Erdelyi [1,2], Kober [7], Sneddon [15], Kalla [10], Saxena [13], Srivastava and Buschmann [16] etc. These operators play an important role in the theory of integral equations and problems concerning Mathematical Physics. In this paper, we shell study the following multidimensional fractional integral operators having the general function  as their kernel. Also, for the sake of brevity, we shall use the symbol f(x) to represent f(x1, x2,., xr). The multivariable I-function introduced by Prasad [12] will be define and represent it in the following manner:


Introduction
Fractional integral operators have been defined and studied by various authors notably by Riemann and Liouville [4], Weyl [4], Erdelyi [1,2], Kober [7], Sneddon [15], Kalla [10], Saxena [13], Srivastava and Buschmann [16] etc. These operators play an important role in the theory of integral equations and problems concerning Mathematical Physics. In this paper, we shell study the following multidimensional fractional integral operators having the general function  as their kernel. Also, for the sake of brevity, we shall use the symbol f(x) to represent f(x 1 , x 2 ,., x r ). The multivariable I-function introduced by Prasad [12] will be define and represent it in the following manner: . Here (i) denotes the numbers of dashes. The contours L i in the complex s i -plane is of the Mellin-Barnes type which runs from -w to +w with indentations, if necessary, to ensure that all the poles of are separated from those of .
For further details and asymptotic expansion of the I-function one can refer by Prasad [12].
In what follows, the multivariable I-function defined by [12] will be represented in the contracted notation: Or simply by I[z 1 ,..., z r ] We introduce the fractional integration operators by means of the following integral equations: Where the kernel  is such that the above integrals make sense. The above operators exist under the following conditions: The following special case of the multidimensional fractional integral operators involving product of Gauss's hypergeometric functions ( [14],p.153,eq. (i) and (ii))will be used in Section 3. (1.6)

Bulletin of Mathematical Sciences and Applications Vol. 11 13
And (1.7) The conditions of existence of these operators follow easily from the conditions given in the paper referred to above. The generalized multidimensional integral transform T, defined below, will also be required during the course of our study: Where k(s 1 ,x 1 ,...,s r ,x r ) is the kernel of the transform T and the multiple integral occurring in the equation (1.8) is assumed to be convergent.
The following multivariable I-function transform will also be used in the sequel: The transform defined above will be denoted symbolically as follows: (1.10)

Relationship Between Multidimensional Fractional Integral operators and Multidimensional Integral Transforms
In this section, we shall establish two most general theorems exhibiting interconnections between the fractional integral operators Y and N defined by (1.4) and (1.5) respectively and the integral transform T defined by (1.8). Next, we give two interesting corollaries interconnecting the multidimensional fractional integral operators defined by (1.6) and (1.7) and the multidimensional I-function transform defined by (1.9).   Where (3.14) Provided that The one and two-dimensional analogous of Theorems 1 and 2 can easily be deduced but since the theorems contain a reasonably detailed analysis of the multidimensional case we prefer to omit their details. The corollaries 1 and 2 given earlier are also new. They give rise to interesting theorems involving multidimensional analogues of fractional integral operators defined by Kober [7], Riemann-Liouville [4] and Weyl [4] and simpler multidimensional integral transforms, on suitably specializing the fractional integral operators and multidimensional I-function transform involved therein. Again, the one and two-dimensional analogues of corollaries 1 and 2, yield theorems essentially similar to those given earlier by Gupta