# §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

18.33.4
18.33.5
18.33.6

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

Assume that . Set

18.33.7

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

## §18.33(iv) Special Cases

### ¶ Askey

When the Askey case is also known as the Rogers–Szegö case.

## §18.33(v) Biorthogonal Polynomials on the Unit Circle

See Baxter (1961) for general theory. See Askey (1982) and Pastro (1985) for special cases extending (18.33.13)–(18.33.14) and (18.33.15)–(18.33.16), respectively. See Gasper (1981) and Hendriksen and van Rossum (1986) for relations with Laurent polynomials orthogonal on the unit circle. See Al-Salam and Ismail (1994) for special biorthogonal rational functions on the unit circle.