# §18.33 Polynomials Orthogonal on the Unit Circle

## §18.33(i) Definition

A system of polynomials , , where is of proper degree , is orthonormal on the unit circle with respect to the weight function () if

where the bar signifies complex conjugate. See Simon (2005a, b) for general theory.

## §18.33(ii) Recurrence Relations

Denote

where , and are constants. Also denote

where the bar again signifies compex conjugate. Then

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## §18.33(iii) Connection with OP’s on the Line

Assume that . Set

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Let and , , be OP’s with weight functions and , respectively, on . Then