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In mathematics, the '''Askey–Wilson polynomials''' (or '''''q''-Wilson polynomials''') are a family of [[orthogonal polynomials]] introduced by {{harvs|txt|last=Askey|author1-link=Richard Askey|last2=Wilson|author2-link=James A. Wilson|year=1985}} as [[q-analog]]s of the [[Wilson polynomials]]. They include many of the other orthogonal polynomials in 1 variable as [[Special case|special]] or [[limiting case (mathematics)|limiting case]]s, 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 ({{math|''C''{{su|b=1|p=&or;}}, ''C''<sub>1</sub>}}), and their 4 parameters {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} correspond to the 4 orbits of roots of this root system.

They are defined by 

:<math>p_n(x) =
p_n(x;a,b,c,d\mid q) :=
a^{-n}(ab,ac,ad;q)_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] </math>

where {{mvar|φ}} is a [[basic hypergeometric function]], {{math|''x'' {{=}} cos &theta;}}, and {{math|(,,,)<sub>''n''</sub>}} is the [[q-Pochhammer symbol|''q''-Pochhammer symbol]]. '''Askey–Wilson functions''' are a generalization to non-integral values of {{mvar|n}}.

== Proof ==

This result can be proven since it is known that

:<math>p_n(\cos{\theta}) = p_n(\cos{\theta};a,b,c,d\mid q)</math>

and using the definition of the ''q''-Pochhammer symbol

:<math>p_n(\cos{\theta})=
a^{-n}\sum_{\ell=0}^{n}q^{\ell}\left(abq^{\ell},acq^{\ell},adq^{\ell};q\right)_{n-\ell}\times\frac{\left(q^{-n},abcdq^{n-1};q\right)_{\ell}}{(q;q)_{\ell}}\prod_{j=0}^{\ell-1}\left(1-2aq^j\cos{\theta}+a^2q^{2j}\right)</math>

which leads to the conclusion that it equals

:<math>a^{-n}(ab,ac,ad;q)_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] </math>

==See also==
*[[Askey scheme]]

==References==
*{{Citation | authorlink=Richard Askey | last1=Askey | first1=Richard | last2=Wilson | first2=James | title=Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials | isbn=978-0-8218-2321-7 | mr=783216 | year=1985 | journal=Memoirs of the American Mathematical Society | issn=0065-9266 | volume=54 | issue=319 | pages=iv+55|url=https://books.google.com/books?id=9q9o03nD_xsC | doi=10.1090/memo/0319}}
*{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | mr=2128719 | year=2004 | volume=96}}
*{{dlmf|id=18.28|title=Askey-Wilson class|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
*{{Citation | first=Tom H. | last=Koornwinder | title=Askey-Wilson polynomial | journal=Scholarpedia | volume=7 | year=2012 | issue=7 | pages=7761 | doi=10.4249/scholarpedia.7761 | bibcode=2012SchpJ...7.7761K | doi-access=free }}

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[[Category:Q-analogs]]
[[Category:Hypergeometric functions]]
[[Category:Orthogonal polynomials]]


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