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 (), and their 4 parameters , , , correspond to the 4 orbits of roots of this root system.
They are defined by
where is a basic hypergeometric function, , and is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of .
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
See also Askey scheme
References