Evaluation of Matrix-variate Gamma and Beta Integrals as Multiple Integrals and Kober Fractional Integral Operators in the Complex Matrix Variate Case

Abstract: Explicit evaluations of matrix-variate gamma and beta integrals in the complex domain by using conventional procedures is extremely difficult. Such an evaluation will reveal the structure of these matrix-variate integrals. In this article, explicit evaluations of matrix-variate gamma and beta integrals in the complex domain for the order of the matrix p = 1; 2 are given. Then fractional integral operators of the Kober type are given for some specific cases of the arbitrary function. A formal definition of fractional integrals in the complex matrix-variate case was given by the author earlier as the M-convolution of products and ratios, where Kober operators become a special class of fractional integral operators.