This paper deals with the interpretation of certain multi-index (multiple) generalized fractional integrals introduced in Kiryakova [4] by using statistical distribution theory and, in particular, the beta distribution and some of its extensions. Then, the fractional integrals are extended to matrix-variate cases in the real and complex domains. Further extensions to rectangular matrix-variate cases in the real and complex domain are also given. It is shown that the approach relying on statistical distribution theory can readily be applied to extend the results to matrix-variate cases which are based on symmetric ratios and symmetric products of matrices whose distribution can also be determined by making use of M-convolutions. Several extensions to the scalar and matrix-variate cases are discussed and a method of obtaining the corresponding differential equation for the corresponding fractional integral is presented.
