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

TL;DR

This paper derives explicit evaluations of complex matrix-variate gamma and beta integrals for small matrix sizes and explores Kober fractional integral operators, revealing their structure and extending fractional calculus to complex matrices.

Contribution

It provides explicit evaluations of matrix-variate gamma and beta integrals in the complex domain for p=1,2, and introduces fractional integral operators of Kober type in this context.

Findings

Explicit evaluations for p=1,2 matrices

Connection of Kober operators to fractional integrals

Structural insights into complex matrix-variate integrals

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.