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

We show that if μ is a probability measure with infinite support on the unit circle having no singular component and a differentiable weight, then the corresponding paraorthogonal polynomial Φn(z;β) solves an explicit second order linear differential equation. We also show that if τ 6= β, then the pair (Φn(z;β),Φn(z; τ)) 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.