EXTENSION OF BETA MATRIX FUNCTION

In the present paper, we give the spectral condition (1:7) on the matrices P;Q and X so that the extension of the Beta matrix function B(P;Q;X) satisfies B(P;Q;X) = Γ(P;X)Γ(Q;X)Γ−1(P +Q;X): Furthermore, some properties of the extension of this function are explained. It is proved that the extension is connected to the Macdonald, error and Whittaker matrix functions and Laguerre matrix polynomials. Applications of this function in (for example) statistics and probability theory are shown.