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.[1] 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 ∣ q ) := a − n ( a b , a c , a d ; q ) n 4 ϕ 3 [ q − n a b c d q n − 1 a e i θ a e − i θ a b a c a d ; q , q ] {\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]}

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 ⁡ θ ) = p n ( cos ⁡ θ ; a , b , c , d ∣ q ) {\displaystyle 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 ⁡ θ ) = a − n ∑ ℓ = 0 n q ℓ ( a b q ℓ , a c q ℓ , a d q ℓ ; q ) n − ℓ × ( q − n , a b c d q n − 1 ; q ) ℓ ( q ; q ) ℓ ∏ j = 0 ℓ − 1 ( 1 − 2 a q j cos ⁡ θ + a 2 q 2 j ) {\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^{2}q^{2j}\right)}

which leads to the conclusion that it equals

a − n ( a b , a c , a d ; q ) n 4 ϕ 3 [ q − n a b c d q n − 1 a e i θ a e − i θ a b a c a d ; q , q ] {\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]}

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, MR 0783216
  • 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, MR 2128719
  • 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.
  • Koornwinder, Tom H. (2012), "Askey-Wilson polynomial", Scholarpedia, 7 (7): 7761, Bibcode:2012SchpJ...7.7761K, doi:10.4249/scholarpedia.7761


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