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 | ||||
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 .
18.33.12 | ||||
18.33.15 | |||
with
18.33.16 | |||
. | |||
For the notation, including the basic hypergeometric function , see §§17.2 and 17.4(i).
When the Askey case is also known as the Rogers–Szegő case.
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.