Erdélyi-Kober fractional integral operators from a statistical perspective -II

In this paper we examine the densities of a product and a ratio of two real positive definite matrix-variate random variables X₁ and X₂, which are statistically independently distributed, and we consider the density of the product U₁=X₂\1/2X₁X₂\1/2 as well as the density of the ratio U₂=X₂\1/2X₁⁻¹X₂\1/2. We define matrix-variate Kober fractional integral operators of the first and second kinds from a statistical perspective, making use of the derivation in the predecessor of this paper for the scalar variable case, by deriving the densities of product cand ratios where one variable has a matrix-variate type-1 beta density and the other matrix variable has an arbitrary density, in the sense, any real-valued scalar function f(X) of matrix argument X, such that f(X) is non-negative for all X and the total integral over all X, on the support of f(X), is unity. A number of generalizations are considered, by using pathway models, by appending matrix variate hypergeometric series etc. During this process matrix-variate Saigo operator and other operators are also defined and properties studied.