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arXiv · 1501.05672

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

Brian Simanek
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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.

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Brian Simanek. 2015-01-22. An Electrostatic Interpretation of the Zeros of Paraorthogonal Polynomials on the Unit Circle. https://arxiv.org/abs/1501.05672

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