Askey–Wilson polynomials

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

$p_n(x;a,b,c,d|q) = (ab,ac,ad;q)_na^{-n}\;_{4}\phi_3 \left[\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta} \\ ab&ac&ad \end{matrix} ; q,q \right]$

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.

Askey–Wilson polynomials are the special case of Koornwinder polynomials (or Macdonald polynomials) for the non-reduced root system of type (C∨
1, C₁).

References

• Askey, Richard; Wilson, James (1985), "Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials", Memoirs of the American Mathematical Society 54 (319): iv+55, ISBN 978-0-8218-2321-7, ISSN 0065-9266, MR783216, http://books.google.com/books?id=9q9o03nD_xsC 
• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR2128719 
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Askey-Wilson class", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248, http://dlmf.nist.gov/18.28