An Electrostatic Interpretation of the Zeros of Paraorthogonal Polynomials on the Unit Circle

TL;DR

This paper demonstrates that zeros of paraorthogonal polynomials on the unit circle can be interpreted as particles in electrostatic equilibrium, derived from explicit differential equations linked to the underlying measure.

Contribution

It establishes a novel electrostatic interpretation of zeros of paraorthogonal polynomials via explicit differential equations, connecting measure properties to particle equilibrium.

Findings

Zeros correspond to electrostatic equilibrium points

Explicit differential equations are derived for these polynomials

The interpretation applies to measures with differentiable weights

Abstract

We show that if m is a probability measure with infinite support on the unit circle having no singular component and a differentiable weight, then the corresponding paraorthogonal polynomial P_n(z;B) solves an explicit second order linear differential equation. We also show that if T and B are distinct, then the pair {P_n(z;B),P_n(z;T)} solves an explicit first order linear system of differential equations. One can use these differential equations to deduce that the zeros of every paraorthogonal polynomial mark the locations of a set of particles that are in electrostatic equilibrium with respect to a particular external field.