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. 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, and their 4 parameters,,, correspond to the 4 orbits of roots of this root system.
They are defined by
pn(x)= pn(x;a,b,c,d\midq):= a-n(ab,ac,ad;q)n 4\phi3\left[\begin{matrix}q-n&abcdqn-1&aei\theta&ae-i\theta\ ab&ac&ad\end{matrix};q,q\right]
where is a basic hypergeometric function,, and is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of .
This result can be proven since it is known that
pn(\cos{\theta})=pn(\cos{\theta};a,b,c,d\midq)
and using the definition of the q-Pochhammer symbol
| -n | |
| p | |
| n(\cos{\theta})= a |
| n | |
| \sum | |
| \ell=0 |
q\ell\left(abq\ell,acq\ell,adq\ell;q\right)n-\ell x
| \left(q-n,abcdqn-1;q\right)\ell | |
| (q;q)\ell |
| \ell-1 | |
| \prod | |
| j=0 |
\left(1-2aqj\cos{\theta}+a2q2j\right)
which leads to the conclusion that it equals
a-n(ab,ac,ad;q)n 4\phi3\left[\begin{matrix}q-n&abcdqn-1&aei\theta&ae-i\theta\ ab&ac&ad\end{matrix} ;q,q\right]