Askey–Wilson polynomials
In mathematics, the Askey–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 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 thatand using the definition of the q-Pochhammer symbol
which leads to the conclusion that it equals