Family of orthogonal polynomials made by Askey & Wilson as q-analogs of Wilson polynomials
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Wikipedia creation date
10/2/2008
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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 p n ( x ; a , b , c , d | q ) = ( a b , a c , a d ; q ) n a − n 4 ϕ 3 [ q − n a b c d q n − 1 a e i θ a e − i θ a b a c a d ; q , q ] {\displaystyle p_{n}(x;a,b,c,d|q)=(ab,ac,ad;q)_{n}a^{-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.
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