Orthogonal polynomials on the unit circle

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 formula_1 is a probability measure on the unit circle in the complex plane, whose support is not finite. The orthogonal polynomials associated to formula_1 are the polynomials formula_3 with leading term formula_4 that are orthogonal with respect to the measure formula_1.
The Szegő recurrence.
Szegő's recurrence states that
where
is the polynomial with its coefficients reversed and complex conjugated, and where the Verblunsky coefficients formula_9 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 formula_10 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 formula_1 form an absolutely convergent series and the weight function formula_13 is strictly positive everywhere.
Szegő's theorem.
Szegő's theorem states that 
where formula_15 is the absolutely continuous part of the measure formula_1.
Rakhmanov's theorem.
Rakhmanov's theorem states that if the absolutely continuous part formula_13 of the measure formula_1 is positive almost everywhere then the Verblunsky coefficients formula_9 tend to 0.
Examples.
The Rogers–Szegő polynomials are an example of orthogonal polynomials on the unit circle.