Askey–Wilson polynomials

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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
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 x = cos(θ) and (,,,)n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of n.

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