Askey–Wilson polynomials

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.^[1] 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

${\displaystyle p_{n}(x)=p_{n}(x;a,b,c,d\mid q):=a^{-n}(ab,ac,ad;q)_{n}\;_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abcdq^{n-1}&ae^{i\theta }&ae^{-i\theta }\\ab&ac&ad\end{matrix}};q,q\right]}$

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

Proof

This result can be proven since it is known that

${\displaystyle p_{n}(\cos {\theta })=p_{n}(\cos {\theta };a,b,c,d\mid q)}$

and using the definition of the q-Pochhammer symbol

${\displaystyle p_{n}(\cos {\theta })=a^{-n}\sum _{\ell =0}^{n}q^{\ell }\left(abq^{\ell },acq^{\ell },adq^{\ell };q\right)_{n-\ell }\times {\frac {\left(q^{-n},abcdq^{n-1};q\right)_{\ell }}{(q;q)_{\ell }}}\prod _{j=0}^{\ell -1}\left(1-2aq^{j}\cos {\theta }+a^{2}q^{2j}\right)}$

which leads to the conclusion that it equals

${\displaystyle a^{-n}(ab,ac,ad;q)_{n}\;_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abcdq^{n-1}&ae^{i\theta }&ae^{-i\theta }\\ab&ac&ad\end{matrix}};q,q\right]}$

See also

• Askey scheme