Total positivity, spherical series, and hypergeometric functions of matrix argument (Q583441)

The paper is concerned with establishing the total positivity of a real function \(K(x,y)=f(xy)\), where f is analytic. With f there is, for each positive integer n, associated a certain spherical series \(\psi_{n,f}\) on a subset of \(S_ n\), the space of Hermitian \(n\times n\) matrices. An integral of \(\psi_{n,f}\) over U(n), the unitary matrices, is derived; next, the connection between the total positivity of K and the non- negativity of the spherical functions is established. In the second part of the paper, the authors treat the total positivity of a classical hypergeometric function \({}_ pF_ q(...,xy)\). It turns out to be equivalent to the non-negativity of a countable set of hypergeometric functions of a matrix variable having, again, p and q parameters. (Incidentally, hypergeometric functions of two matrix variables are also considered. They are not analogues of classical double hypergeometric functions.) Finally, the Gauss and Kummer cases are studied in greater detail.