Total positivity of a Cauchy kernel

We study the total positivity of the kernel 1/(x2+2cos(πα)xy+y2)1/(x2+2cos(πα)xy+y2) over R+×R+R+×R+. The case of infinite order is characterized by an application of Schoenberg’s theorem. We give necessary conditions for the case of finite order with the help of Chebyshev polynomials of the second kind. Sufficient conditions for the case of finite order are obtained thanks to the Izergin–Korepin determinant and its expression in terms of alternating sign matrices. A conjecture on the generating function of alternating sign matrices with a fixed number of negative entries arises in a natural way from our study. As a by-product, we give a partial answer to a question of Karlin on positive stable semigroups.