Askey–Wilson polynomials

In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by Richard Askey and James A. Wilson as q-analogs of the Wilson polynomials.{{sfnp|Askey|Wilson|1985}} 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 ({{math|C{{su|b=1|p=∨}}, C₁}}), 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

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

where {{mvar|φ}} is a basic hypergeometric function, {{math|x {{=}} cos θ}}, and {{math|(,,,)_n}} is the 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

: $p_n(\cos{\theta}) = p_n(\cos{\theta};a,b,c,d\mid q)$

and using the definition of the q-Pochhammer symbol

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

which leads to the conclusion that it equals

: $a^{-n}(ab,ac,ad;q)_n\;_{4}\phi_3 \left[\begin{matrix}
q^{-n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta} \\
ab&ac&ad \end{matrix}
; q,q \right]$

References

{{Citation | author-link1=Richard Askey | last1=Askey | first1=Richard | author-link2=James A. Wilson | 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 }}

Category:Q-analogs

Category:Hypergeometric functions

Category:Orthogonal polynomials

Askey–Wilson polynomials

Proof

See also

References