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 14). See
also Gasper and Rahman (2004, pp. 180–199),
Ismail (2009, Chapter 15), and Koornwinder (2012).
For the notation of -hypergeometric
functions see §§17.2 and 17.4(i).