Askey–Wilson polynomials

In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by Askey and Wilson (1985) as q-analogs of the Wilson polynomials. 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 ( $C \lor 1, C 1$ ), 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 $n$ .

Proof

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