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

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.