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

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We show that if $\mu$ is a probability measure with infinite support on the unit circle having no singular component and a differentiable weight, then the corresponding paraorthogonal polynomial $\Phi_n(z;\beta)$ solves an explicit second order linear differential equation. We also show that if $\tau\neq\beta$, then the pair $(\Phi_n(z;\beta),\Phi_n(z;\tau))$ 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.