[1] They include many of the other orthogonal polynomials in 1 variable as special or limiting cases, described in the Askey scheme.
Askey–Wilson polynomials are the special case of Macdonald polynomials (or Koornwinder polynomials) for the non-reduced affine root system of type (C∨1, C1), and their 4 parameters a, b, c, d correspond to the 4 orbits of roots of this root system.
They are defined by where φ is a basic hypergeometric function, x = cos θ, and (,,,)n is the q-Pochhammer symbol.
Askey–Wilson functions are a generalization to non-integral values of n. This result can be proven since it is known that and using the definition of the q-Pochhammer symbol which leads to the conclusion that it equals
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