Total positivity, spherical series, and hypergeometric functions of matrix argument

Given a totally positive function K of two real variables, is there a method for establishing the total positivity of K in an "obvious" fashion? In the case in which K(x, y) = f(xy), where f is real-analytic in a neighborhood of zero, we obtain integral representations for the determinants which define the total positivity of K. The total positivity of K then follows immediately from positivity of the integrands. In particular, we analyze the total positivity of classical hypergeometric functions by these methods. The central theme of this work is the circle of ideas that relates total positivity to "spherical series" on the symmetric space GL(n, C) U(n), and classical hypergeometric functions to hypergeometric functions of matrix argument.