The Askey–Wilson class OP’s comprise the four-parameter families of Askey–Wilson polynomials and of -Racah polynomials, and cases of these families obtained by specialization of parameters. The Askey–Wilson polynomials form a system of OP’s , , that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. The -Racah polynomials form a system of OP’s , , that are orthogonal with respect to a weight function on a sequence , , with a constant. Both the Askey–Wilson polynomials and the -Racah polynomials can best be described as functions of (resp. ) such that in the Askey–Wilson case, and in the -Racah case, and both are eigenfunctions of a second-order -difference operator similar to (18.27.1).
In the remainder of this section the Askey–Wilson class OP’s are defined by their -hypergeometric representations, followed by their orthogonal properties. For further properties see Koekoek et al. (2010, Chapter 3). See also Gasper and Rahman (2004, pp. 180–199) and Ismail (2005, Chapter 15). For the notation of -hypergeometric functions see §§17.2 and 17.4(i).
Assume are all real, or two of them are real and two form a conjugate pair, or none of them are real but they form two conjugate pairs. Furthermore, , , , , , . Then
Eq. (18.28.10) is valid when either
If, in addition to (18.28.11) or (18.28.12), we have , then the measure in (18.28.10) is uniquely determined. Also, if , then (18.28.10) holds with interchanged. For further nondegenerate cases see Chihara and Ismail (1993) and Christiansen and Ismail (2006).
These polynomials are also called Rogers polynomials.
For and see Koekoek et al. (2010, Eq. (3.2.2)).