Askey–Wilson polynomials

In mathematics, the Askey-Wilson polynomials are the polynomials

$p_{n}(x;a,b,c,d|q)=(ab,ac,ad;q)_{n}a^{-n}\;_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abcdq^{n-1}&ae^{i\theta }&ae^{-i\theta }\\ab&ac&ad\end{matrix}};q,q\right]$

where φ is a basic hypergeometric function and x = cos(θ) and (,,,)_n is the q-Pochhammer symbol.

They were introduced by Askey & Wilson (1985) as q-analogues of the Wilson polynomials.

Askey–Wilson polynomials are he specail case of 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
• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 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.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.