Hypergeometric functions of many matrix variables and distributions of generalized quadratic forms

A representation of the exact density of a linear function of independent matrix-variate gamma variables, in terms of hypergeometric function of many matrix variables, is considered. The result is obtained in terms of a ø2 function of many matrix variables. Connection of the linear function to generalized quadratic forms is pointed out. The exact moments and the exact density of the determinant of such a linear combination are also derived.