§18.33 Polynomials Orthogonal on the Unit Circle
Contents
- §18.33(i) Definition
- §18.33(ii) Recurrence Relations
- §18.33(iii) Connection with OP’s on the Line
- §18.33(iv) Special Cases
- §18.33(v) Biorthogonal Polynomials on the Unit Circle
§18.33(i) Definition
§18.33(ii) Recurrence Relations
Denote
18.33.2
where
, and
are constants. Also
denote
18.33.3
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
18.33.8
18.33.9
Conversely,
18.33.10
18.33.11
where
,
,
, and
are independent of
.
§18.33(iv) Special Cases
¶ Trivial
18.33.12
§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.



