In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane, for some probability measure on the unit circle. They were introduced by . Definition Suppose that is a probability measure on the unit circle in the complex plane, whose support is not finite. The orthogonal polynomials associated to are the polynomials with leading term that are orthogonal with respect to the measure . The Szegő recurrence Szegő's recurrence states that where is the polynomial with its coefficients reversed and complex conjugated, and where the Verblunsky coefficients are complex numbers with absolute values less than 1. Verblunsky's theorem Verblunsky's theorem states that any sequence of complex numbers in the open unit disk is the sequence of Verblunsky coefficients for a unique probability measure on the unit circle with infinite support. Geronimus's theorem Geronimus's theorem states that the Verblunsky coefficients of the measure μ are the Schur parameters of the function defined by the equations Baxter's theorem Baxter's theorem states that the Verblunsky coefficients form an absolutely convergent series if and only if the moments of form an absolutely convergent series and the weight function is strictly positive everywhere. Szegő's theorem Szegő's theorem states that where is the absolutely continuous part of the measure . Rakhmanov's theorem Rakhmanov's theorem states that if the absolutely continuous part of the measure is positive almost everywhere then the Verblunsky coefficients tend to 0. Examples The Rogers–Szegő polynomials are an example of orthogonal polynomials on the unit circle. References Orthogonal polynomials