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

$p_n(x)=p_n(x;a,b,c,d\mid q):=a^{-n}(ab,ac,ad;q{)}_n{}_4{\varphi }_3[\begin{array}{cccc}{q}^{-n}& abcd{q}^{n-1}& a{e}^{i\theta }& a{e}^{-i\theta }\\ ab& ac& ad\end{array};q,q]$

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

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

and using the definition of the q-Pochhammer symbol

$p_n(\cos \theta )=a^{-n}\sum _{\ell =0}^n{q}^{\ell }{(ab{q}^{\ell },ac{q}^{\ell },ad{q}^{\ell };q)}_{n-\ell }\times \frac{{(q^{-n},abcd{q}^{n-1};q)}_{\ell }}{(q;q{)}_{\ell }}\prod _{j=0}^{\ell -1}(1-2a{q}^j\cos \theta +a^2{q}^{2j})$

which leads to the conclusion that it equals

$a^{-n}(ab,ac,ad;q{)}_n{}_4{\varphi }_3[\begin{array}{cccc}{q}^{-n}& abcd{q}^{n-1}& a{e}^{i\theta }& a{e}^{-i\theta }\\ ab& ac& ad\end{array};q,q]$

See also

• Askey scheme

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