Concept information
Preferred term
Askey-Wilson polynomial
Definition(s)
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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, x = cos θ, and (,,,)n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of n.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Askey%E2%80%93Wilson_polynomials)
![{\\displaystyle p_{n}(x)=p_{n}(x;a,b,c,d\\mid q):=a^{-n}(ab,ac,ad;q)_{n}\\;_{4}\\phi _{3}\\left[{\egin{matrix}q^{-n}&abcdq^{n-1}&ae^{i\ heta }&ae^{-i\ heta }\\\\ab&ac&ad\\end{matrix}};q,q\
ight]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0fd99eb1abe68fe7e32885d60a8ba10f11db65)
Broader concept(s)
Synonym(s)
- q-Wilson polynomial
In other languages
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French
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q-polynôme de Wilson
URI
http://data.loterre.fr/ark:/67375/PSR-J4S0TSB9-W
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