Slater determinants of orthogonal polynomials

TL;DR

This paper explores the properties of Slater determinants of orthogonal polynomials, revealing their connections to Hankel determinants, Selberg integrals, and applications in positivity and Jensen polynomials.

Contribution

It establishes new representations of symmetrized Slater determinants using Hankel determinants and integrals, expanding understanding of their structure and applications.

Findings

Representation of Slater determinants via Hankel determinants

Connections to Selberg type integrals

Applications to positive polynomials and Jensen polynomials

Abstract

The symmetrized Slater determinants of orthogonal polynomials with respect to a non-negative Borel measure are shown to be represented by constant multiple of Hankel determinants of two other families of polynomials, and they can also be written in terms of Selberg type integrals. Applications include positive determinants of polynomials of several variables and Jensen polynomials and its derivatives for entire functions.