Askey–Wilson polynomials
In mathematics, the Askey–Wilson polynomials are a family of orthogonal polynomials introduced by 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 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 x = cos and n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of n.
