Askey-Wilson polynomial

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Author: Dr. Tom H. Koornwinder, Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands

Dr. Tom Koornwinder accepted the invitation on 23 September 2008 (self-imposed deadline: 6 May 2009).

Askey-Wilson polynomial refers to a four-parameter family of q-hypergeometric orthogonal polynomials which contains all families of classical orthogonal polynomials (in the wide sense) as special or limit cases.

Contents

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Orthogonal Polynomials

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General Orthogonal Polynomials

Let w(x) be a nonnegative function on an open real interval (a,b) such that the integral \int_a^b|x|^n w(x)\,dx is well-defined and finite for all nonnegative integers n. A system of real-valued polynomials p_n(x) (n=0,1,2,\ldots) is called orthogonal on the interval (a,b) with respect to the weight function w(x) if p_n(x) has degree n and if \int_a^b p_n(x) p_m(x) w(x)\,dx=0 for n\ne m.

More generally we can replace in this definition w(x)\,dx by a positive measure d\mu(x) on \Bbb R. Then the orthogonality relation becomes \int_{\Bbb R} p_n(x) p_m(x)\,d\mu(x)=0\quad(n\ne m). If the measure is discrete then this takes the form \sum_{j=1}^\infty p_n(x_j) p_m(x_j)\,w_j=0\quad(n\ne m), where the weights w_j are positive.

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Classical Orthogonal Polynomials (in the strict sense)

A system of orthogonal polynomials p_n(x) is called classical (in the strict sense) if there is a second order linear differential operator L, not depending on n, such that p_n is an eigenfunction of L for each n:

(1)
L p_n=\lambda_n p_n.

Examples of classical orthogonal polynomials are the Legendre polynomials P_n(x) (w(x):=1, (a,b):=(-1,1)) and the Hermite polynomials H_n(x) (w(x):=e^{-x^2}, (a,b):=(-\infty,\infty)).

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Classical Orthogonal Polynomials (in the wide sense)

More generally, a system of orthogonal polynomials p_n(x) is called classical (in the wide sense) if there is a second order linear difference or q\mbox{-}difference operator L, not depending on n, such that (1) holds.

  • Difference operator, for instance (Lf)(x):=a(x) f(x-1)+b(x) f(x)+c(x) f(x+1). An example are the Charlier polynomials C_n(x;a) (a>0) which are orthogonal with respect to the weights a^x/x! on the points x (x=0,1,2,\ldots).
  • q\mbox{-}Difference operator, for instance (Lf)(x):=a(x) f(q^{-1}x)+b(x) f(x)+c(x) f(qx). An example are the discrete q\mbox{-}Hermite I polynomials h_n(x;q) which are orthogonal with respect to the weights q^j\prod_{k=1}^\infty(1-q^{2k+2j+2}) on the points \pm q^j (j=0,1,2,\ldots).
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Hypergeometric and Basic Hypergeometric Series

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Hypergeometric Series

For complex a and nonnegative integer n let (a)_n:=a(a+1)\ldots(a+n-1) be the Pochhammer symbol.

A hypergeometric series with r upper parameters a_1,\ldots,a_r and s lower parameters b_1,\ldots,b_s is formally defined as

(2)
{}_rF_s\!\left(\begin{matrix}{a_1,\ldots,a_r}\\{b_1,\ldots,b_s}\end{matrix};z\right):=\sum_{k=0}^\infty \frac{(a_1)_k\cdot\cdot\cdot(a_r)_k}{(b_1)_k\cdot\cdot\cdot(b_s)_k}\,\frac{z^k}{k!}.

If a_1 is equal to a nonpositive integer -n then the series on the right-hand side of (2) terminates after the term with k=n.

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Basic Hypergeometric Series

Let q be a complex number not equal to 0 or 1.

For complex a and nonnegative integer n let (a;q)_n:=(1-a)(1-aq)\ldots(1-aq^{n-1}) be the q\mbox{-}Pochhammer symbol.

Also let (a_1,a_2,\ldots,a_r;q)_n:=(a_1;q)_n (a_2;q)_n\ldots(a_r;q)_n.

For |q|<1 let (a;q)_\infty:=\prod_{k=0}^\infty (1-aq^k), a convergent infinite product.

A basic or q\mbox{-}hypergeometric series with r upper parameters a_1,\ldots,a_r and s lower parameters b_1,\ldots,b_s is formally defined as

(3)
{}_r\phi_s\!\left(\begin{matrix}{a_1,\ldots,a_r}\\{b_1,\ldots,b_s}\end{matrix};q,z\right):=\sum_{k=0}^\infty \bigl((-1)^k q^{k(k-1)/2}\bigr)^{s-r+1}\,\frac{(a_1,\ldots,a_r;q)_k}{(b_1,\ldots.b_s;q)_k}\,\frac{z^k}{(q;q)_k}.

If a_1=q^{-n} for a nonnegative integer n then the series on the right-hand side of (3) terminates after the term with k=n.

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Askey-Wilson Polynomials

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Definition

(4)
p_n(\cos\theta)=p_n(\cos\theta;a,b,c,d\mid q):=\frac{(ab,ac,ad;q)_n}{a^n}\,{}_4\phi_3\!\left(\begin{matrix}{q^{-n},q^{n-1}abcd,ae^{i\theta},ae^{-i\theta}}\\{ab,ac,ad}\end{matrix};q,q\right)
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Orthogonality relation

Let 0<q<1. Assume that a,b,c,d are four reals, or two reals and one pair of complex conjugates, or two pairs of complex conjugates. Also assume that |a|,|b|,|c|,|d|<1. Then

(5)
\int_{-1}^1 p_n(x) p_m(x) w(x)\,dx=h_n\,\delta_{n,m},

where

2\pi\sin\theta\,w(\cos\theta):= \left|\frac{(e^{2i\theta};q)_\infty} {(ae^{i\theta},be^{i\theta},ce^{i\theta},de^{i\theta};q)_\infty}\right|^2,

and

h_0:=\frac{(abcd;q)_\infty}{(q,ab,ac,ad,bc,bd,cd;q)_\infty}\,,\quad \frac{h_n}{h_0}:=\frac{1-abcdq^{n-1}}{1-abcdq^{2n-1}}\, \frac{(q,ab,ac,ad,bc,bd,cd;q)_n}{(abcd;q)_n}.

For more general parameter values the orthogonality relation (5) can be given as the contour integral

(6)
\frac1{2\pi i} \oint_C p_n\bigl((z+z^{-1})/2\bigr)\,p_m\bigl((z+z^{-1})/2\bigr)\, \frac{(z^2,z^{-2};q)_\infty} {(az,az^{-1},bz,bz^{-1},cz,cz^{-1},dz,dz^{-1};q)_\infty}\,\frac{dz}z =2h_n\delta_{n,m},

where C is the unit circle traversed in positive direction with suitable deformations to separate the sequences of poles converging to zero from the sequences of poles diverging to \infty. The case n=m=0 of (5) or (6) is called the Askey-Wilson integral.

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q-Difference equation

(7)
A(z)P_n(qz)-\bigl(A(z)+A(z^{-1})\bigr)P_n(z)+A(z^{-1})P_n(q^{-1}z)= (q^{-n}-1)(1-q^{n-1}abcd)P_n(z),

where P_n(z):=p_n\bigl((z+z^{-1})/2\bigr) and A(z):=(1-az)(1-bz)(1-cz)(1-dz)/\bigl((1-z^2)(1-qz^2)\bigr).

By (7) the Askey-Wilson polynomials P_n(z) are eigenfunctions of a second order q\mbox{-}difference operator. Thus they are classical orthogonal polynomials in the wide sense.

Suggested by: Dr. Richard A. Askey, Department of Mathematics, University of Wisconsin-Madison, WI
Invited by: Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
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