Total positivity of a Cauchy kernel

2013/05/06 by Thomas Simon, Simon, Thomas
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR) #math.CA #math.PR

paper · pdf · doi:10.48550/arxiv.1305.1173

arxiv created 2013/05/06 · arxiv updated 2013/05/07

Abstract

We study the total positivity of the kernel 1/(x² + 2 cos(π\a)xy +y²). The case of infinite order is characterized by an application of Schoenberg's theorem. We then give necessary conditions for the cases of any given finite order with the help of Chebyshev polynomials of the second kind. Sufficient conditions for the finite order cases are also obtained, thanks to Propp's formula for the Izergin-Korepin determinant. As a by-product, we give a partial answer to a question of Karlin on positive stable semi-groups.

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