Hypergeometric Functions of Scalar Matrix Argument are Expressible in Terms of Classical Hypergeometric Functions

It is shown that the hypergeometric function of m m scalar matrix argument (cf. Herz, Annals of Math. , 61 (1955), pp. 474–523) may be expressed as the Pfaffian of a matrix whose entries are evaluated in terms of classical hypergeometric functions. Applications, in multivariate statistical theory, are made to the distributions of eigenvalues of various random matrices.