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
- [math]\displaystyle{ 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 φ is a basic hypergeometric function, x = cos θ, and (,,,)n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of n.
Proof
This result can be proven since it is known that
- [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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
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, doi:10.1090/memo/0319, ISBN 978-0-8218-2321-7, ISSN 0065-9266, https://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
- 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 978-0-521-19225-5, http://dlmf.nist.gov/18.28
- Koornwinder, Tom H. (2012), "Askey-Wilson polynomial", Scholarpedia 7 (7): 7761, doi:10.4249/scholarpedia.7761, Bibcode: 2012SchpJ...7.7761K
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