We use classical Jacobi polynomials to identify the equilibrium configurations of charged particles confined to the unit circle. Our main result unifies two theorems from a 1986 paper of Forrester and Rogers.
Spectral quantization of discrete random walks on half-line, and orthogonal polynomials on the unit circle
Discrete Painlevé equations, Orthogonal Polynomials on the Unit Circle and $N$-recurrences for averages over U(N) -- \PVI $τ$-functions
Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle
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