Askey–Wilson polynomials

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

$p_n(x;a,b,c,d|q) = (ab,ac,ad;q)_na^{-n}\;_{4}\phi_3 \left[\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta} \\ ab&ac&ad \end{matrix} ; q,q \right]$

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.

See also

• Askey scheme