Total positivity of a Cauchy kernel

TL;DR

This paper investigates the total positivity properties of a specific Cauchy kernel, providing characterizations for infinite and finite orders, and addressing related questions in stable semi-group theory.

Contribution

It offers new criteria for total positivity of the kernel, applying Schoenberg's theorem, Chebyshev polynomials, and Propp's formula, and partially answers Karlin's question on stable semi-groups.

Findings

Characterization of infinite order total positivity using Schoenberg's theorem.

Necessary and sufficient conditions for finite order total positivity.

Partial resolution of Karlin's question on positive stable semi-groups.

Abstract

We study the total positivity of the kernel $1/(x^2 + 2 \cos(\pi\a)xy +y^2).$ 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.