Erdélyi-Kober Fractional Integrals Involving Many Real Matrices

Abstract

All the matrices appearing in this chapter are p × p real positive definite unless stated otherwise. In order to avoid too many symbols we will use $u_1=\frac {x_2}{x_1}$ for the ratio of x ₂ to x ₁ in the real scalar variable case, $U_1=X_2^{\frac {1}{2}}X_1^{-1}X_2^{\frac {1}{2}}$, symmetric ratio, in the real p × p matrix-variate case. The corresponding density of u ₁ and U ₁ will be indicated by g ₁; we will use u ₂ = x ₁x ₂ for the product in the real scalar variable case and $U_2=X_2^{\frac {1}{2}}X_1X_2^{\frac {1}{2}}$, the symmetric product, in the real p × p matrix-variate case. The corresponding density of u ₂ or U ₂ will be indicated by g ₂. If x ₁ and x ₂ are statistically independently distributed real scalar random variables, and X ₁ and X ₂ are statistically independently distributed real matrix-variate random variables, then g ₂(u ₂) or g ₂(U ₂) and g ₁(u ₁) or g ₁(U ₁) will denote product and ratio distributions or M-convolutions of product and ratio whatever be the set of variables. In all the preceding chapters the basic claim is that fractional integrals are of two kinds, the first kind or left-sided and the second kind or right-sided. The first kind of fractional integrals belong to the class of Mellin convolution of a ratio and the second kind of fractional integrals belong to the class of Mellin convolution of a product. In the matrix-variate case these will be M-convolutions of ratio and product respectively. If the variables are random variables then g ₁ and g ₂ correspond to the densities of ratio and product respectively. We will give the following formal definition of fractional integral operators of the first kind and second kind. For the sake of completeness we will recall the general definitions from Chaps. 3 and 4.