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. 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

$p_n(x)=p_n(x;a,b,c,d\mid q):=a^{-n}(ab,ac,ad;q{)}_n{}_4{\varphi }_3[\begin{array}{cccc}{q}^{-n}& abcd{q}^{n-1}& a{e}^{i\theta }& a{e}^{-i\theta }\\ ab& ac& ad\end{array};q,q]$

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

$p_n(\cos \theta )=p_n(\cos \theta ;a,b,c,d\mid q)$

and using the definition of the q-Pochhammer symbol

$p_n(\cos \theta )=a^{-n}\sum _{\ell =0}^n{q}^{\ell }{(ab{q}^{\ell },ac{q}^{\ell },ad{q}^{\ell };q)}_{n-\ell }\times \frac{{(q^{-n},abcd{q}^{n-1};q)}_{\ell }}{(q;q{)}_{\ell }}\prod _{j=0}^{\ell -1}(1-2a{q}^j\cos \theta +a^2{q}^{2j})$

which leads to the conclusion that it equals

$a^{-n}(ab,ac,ad;q{)}_n{}_4{\varphi }_3[\begin{array}{cccc}{q}^{-n}& abcd{q}^{n-1}& a{e}^{i\theta }& a{e}^{-i\theta }\\ ab& ac& ad\end{array};q,q]$

See also

• Askey scheme

References

• .