In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by Richard Askey and James A. Wilson 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∨ 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, , 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