Multivariate circular Jacobi polynomials

TL;DR

This paper introduces a new multivariate orthogonal polynomial as a deformation of spherical polynomials, connecting harmonic analysis on symmetric cones with circular Jacobi ensembles, and explores its key properties.

Contribution

It presents a novel multivariate polynomial generalizing circular Jacobi polynomials, with detailed analysis of its properties and connections to harmonic analysis and random matrix ensembles.

Findings

Introduces a 2-parameter deformed multivariate polynomial.

Establishes the polynomial's orthogonality and related properties.

Connects the polynomial to circular Jacobi ensembles.

Abstract

We introduce a new multivariate orthogonal polynomial which is a 2-parameter deformation of the spherical polynomial by harmonic analysis on symmetric cone. This is also regarded as a multivariate analogue of the circular Jacobi polynomial. Further, the weight function of its orthogonality relation coincides with the circular Jacobi ensemble defined by Bourgade et al.. We also obtain its main properties : generating function, pseudo differential equation and deteraminant expression.