Askey-Wilson polynomial
Dr. Tom Koornwinder accepted the invitation on 23 September 2008 (self-imposed deadline: 31 January 2011).
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.
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Orthogonal polynomials
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 \(\sum_{j=0}^\infty p_n(x_j) p_m(x_j)\,w_j=0\quad(n\ne m)\ ,\) where the weights \(w_j\) are positive. The finite case \(\sum_{j=0}^N p_n(x_j) p_m(x_j)\,w_j=0\quad(n\ne m;\;n,m=0,1,\ldots,N)\) also occurs.
Three-term recurrence relation
Any system of orthogonal polynomials \(p_n(x)\) satisfies a three-term recurrence relation of the form \[x p_n(x)=A_n p_{n+1}(x)+B_n p_n(x)+C_n p_{n-1}(x),\] where \(p_{-1}(x):=0\) and \(A_{n-1}C_n>0\ .\)
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 \(n\ :\) \[\tag{1} L p_n=\lambda_n p_n.\]
There are three families of orthogonal polynomials which are classical in the strict sense:
- Jacobi polynomials \(P_n^{(\alpha,\beta)}(x)\ ,\) where \(\alpha,\beta>-1\ ,\) \(w(x):=(1-x)^\alpha(1+x)^\beta\ ,\) \((a,b):=(-1,1)\ ;\)
- Laguerre polynomials \(L_n^\alpha(x)\ .\) where \(\alpha>-1\ ,\) \(w(x):=e^{-x} x^\alpha\ ,\) \((a,b):=(0,\infty)\ ;\)
- Hermite polynomials \(H_n(x)\ ,\) where \(w(x):=e^{-x^2}\ ,\) \((a,b):=(-\infty,\infty)\ .\)
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 \(\pm q^j\) (\(j=0,1,2,\ldots\)).
Hypergeometric and basic hypergeometric Series
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 \[\tag{2} {}_rF_s\!\left(\begin{matrix}{a_1,\ldots,a_r}\\{b_1,\ldots,b_s}\end{matrix};z\right):=\sum_{k=0}^\infty \frac{(a_1)_k\cdot\cdot\cdot(a_r)_k}{(b_1)_k\cdot\cdot\cdot(b_s)_k}\,\frac{z^k}{k!}. \]
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 \(k=n\ .\)
Basic hypergeometric series
See Gasper & Rahman [GR]. 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 \[\tag{3} {}_r\phi_s\!\left(\begin{matrix}{a_1,\ldots,a_r}\\{b_1,\ldots,b_s}\end{matrix};q,z\right):=\sum_{k=0}^\infty \bigl((-1)^k q^{k(k-1)/2}\bigr)^{s-r+1}\,\frac{(a_1,\ldots,a_r;q)_k}{(b_1,\ldots.b_s;q)_k}\,\frac{z^k}{(q;q)_k}. \]
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 \(k=n\ .\)
Askey-Wilson polynomials
Askey-Wilson polynomials were introduced by Askey & Wilson [AW] in 1985.
Definition
\[\tag{4} p_n(\cos\theta)=p_n(\cos\theta;a,b,c,d\mid q):=\frac{(ab,ac,ad;q)_n}{a^n}\,{}_4\phi_3\!\left(\begin{matrix}{q^{-n},q^{n-1}abcd,ae^{i\theta},ae^{-i\theta}}\\{ab,ac,ad}\end{matrix};q,q\right).\]
This is a polynomial of degree \(n\) in \(\cos\theta\ .\)
Symmetry
The polynomials \(p_n(x;a,b,c,d\mid q)\) are symmetric in the parameters \(a,b,c,d\ .\)
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 \[\tag{5} \int_{-1}^1 p_n(x) p_m(x) w(x)\,dx=h_n\,\delta_{n,m}, \]
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 \[\tag{6} \frac1{2\pi i} \oint_C p_n\bigl((z+z^{-1})/2\bigr)\,p_m\bigl((z+z^{-1})/2\bigr)\, \frac{(z^2,z^{-2};q)_\infty} {(az,az^{-1},bz,bz^{-1},cz,cz^{-1},dz,dz^{-1};q)_\infty}\,\frac{dz}z =2h_n\delta_{n,m}, \]
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 left-hand side of (6) can be rewritten as the left-hand side of (5) with finitely many terms added of the form \(p_n(x_j) p_m(x_j) w_j\ ,\) where \(x_j\) is in \(\Bbb R\) outside \([-1,1]\ .\) The case \(n=m=0\) of (5) or (6) is called the Askey-Wilson integral.
q-Difference equation
\[\tag{7} A(z)P_n(qz)-\bigl(A(z)+A(z^{-1})\bigr)P_n(z)+A(z^{-1})P_n(q^{-1}z)= (q^{-n}-1)(1-q^{n-1}abcd)P_n(z), \]
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.
Discretization, specializations and limit cases
See Chapter 14 in the book by Koekoek et al. [KLS] or see the earlier online Koekoek & Swarttouw report [KS].
q-Racah polynomials
The \(q\mbox{-}\)Racah polynomials \(R_n(x;\alpha,\beta,\gamma,\delta\mid q)\) form a family of finite (\(n=0,1,\ldots,N\)) sytems of orthogonal polynomials depending on four parameters \(\alpha,\beta,\gamma,\delta\ ,\) where \(q\alpha=q^{-N}\) or \(q\beta\delta=q^{-N}\) or \(q\gamma=q^{-N}\ .\) They have essentially the same analytic expression as the Askey-Wilson polynomials: \[R_n(q^{-y}+\gamma\delta q^{y+1};\alpha,\beta,\gamma,\delta\mid q):={}_4\phi_3\!\left(\begin{matrix}{q^{-n},q^{n+1}\alpha\beta,q^{-y},\gamma\delta q^{y+1}}\\{q\alpha,q\beta\delta,q\gamma}\end{matrix};q,q\right).\] They satisfy an orthogonality relation of the form \[\sum_{y=0}^N R_n(q^{-y}+\gamma\delta q^{y+1}) R_m(q^{-y}+\gamma\delta q^{y+1})\,w_y=h_n\delta_{n,m}\quad(n,m=0,1,\ldots,N).\]
Selected special cases
We obtain special subfamilies of the Askey-Wilson polynomials by specialization of parameters.
- Al-Salam-Chihara polynomials: \(Q_n(x;a,b\mid q):=p_n(x;a,b,0,0\mid q)\ .\)
- Continuous \(q\)-Jacobi polynomials: \(P_n^{(\alpha,\beta)}(x;q):={\rm const.}\, p_n(x;q^{\frac12}, q^{\alpha+\frac12},-q^{\beta+\frac12},-q^{\frac12}\mid q)={\rm const.}\,p_n(x;q^{\alpha+\frac12},q^{\alpha+\frac32}, -q^{\beta+\frac12},-q^{\beta+\frac32}\mid q^2)\ .\)
- Continuous \(q\)-ultraspherical polynomials: \(C_n(\cos\theta;\beta\mid q):= \frac{(\beta;q)_n}{(q;q)_n}\, p_n(\cos\theta;\beta^{\frac12},\beta^{\frac12}q^{\frac12},-\beta^{\frac12},-\beta^{\frac12}q^{\frac12}\mid q)= \sum_{k=0}^n\frac{(\beta;q)_k(\beta;q)_{n-k}}{(q;q)_k(q;q)_{n-k}}\,e^{i(n-2k)\theta}\ .\)
- Continuous \(q\)-Hermite polynomials: \(H_n(\cos\theta\mid q):=(q;q)_n\,C_n(\cos\theta;0\mid q)= \sum_{k=0}^n\frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}\,e^{i(n-2k)\theta}\ .\)
- Chebyshev polynomials: \(p_n(\cos\theta;1,-1,q^{\frac12},-q^{\frac12}\mid q)={\rm const.}\,\cos n\theta,\quad p_n(\cos\theta;q,-q,q^{\frac12},-q^{\frac12}\mid q)={\rm const.}\,\frac{\sin(n+1)\theta}{\sin\theta}\,,\)\(p_n(\cos\theta;q,-1,q^{\frac12},-q^{\frac12}\mid q)={\rm const.}\,\frac{\sin(n+\frac12)\theta}{\sin\frac12\theta}\,,\quad p_n(\cos\theta;1,-q,q^{\frac12},-q^{\frac12}\mid q)={\rm const.}\,\frac{\cos(n+\frac12)\theta}{\cos\frac12\theta}\,.\)
Selected limit cases preserving q
- Big \(q\)-Jacobi polynomials: \(P_n(x;a,b,c;q):={}_3\phi_2\!\left(\begin{matrix}{q^{-n},q^{n+1}ab,x}\\{qa,qc}\end{matrix};q,q\right)={\rm const.}\,\lim_{\lambda\downarrow0}\lambda^n p_n(\tfrac12\lambda^{-1}x;\lambda,\lambda^{-1}qa,\lambda^{-1}qc,\lambda bc^{-1}\mid q).\)
- Little \(q\)-Jacobi polynomials: \(p_n(x;a,b;q):={}_2\phi_1\!\left(\begin{matrix}{q^{-n},q^{n+1}ab}\\{qa}\end{matrix};q,qx\right)={\rm const.}\,\lim_{\lambda\downarrow0}\lambda^n p_n(\tfrac12\lambda^{-1}x;-q^{\frac12}a,qb\lambda,-q^{\frac12},\lambda^{-1}\mid q).\)
Selected limit cases for q to 1
- Wilson polynomials: \(W_n(y^2;a,b,c,d)=\lim_{q\uparrow1}(1-q)^{-3n} p_n(\tfrac12(q^{iy}+q^{-iy});q^a,q^b,q^c,q^d\mid q).\)
- Jacobi polynomials: \(P_n^{(\alpha,\beta)}(x)=\lim_{q\uparrow1}P_n^{(\alpha,\beta)}(x;q).\)
- Ultraspherical polynomials: \(C_n^\lambda(x)=\lim_{q\uparrow1}C_n(x;q^\lambda\mid q).\)
- Hermite polynomials: \(H_n(x)=\lim_{q\uparrow1}(1-q)^{-n/2} H_n((1-q)^{1/2} x\mid q^2).\)
A limit case for q to 0
- Special Bernstein-Szegö polynomials: \(\lim_{q\downarrow0}C_n(\cos\theta;\beta\mid q)=(1-\beta)\,\frac{\sin(n+1)\theta}{\sin\theta}-\beta(1-\beta)\,\frac{\sin(n-1)\theta}{\sin\theta}\quad(n=1,2\ldots) \quad\mbox{and}\quad=1\quad(n=0).\)
Analogues in several variables
Macdonald polynomials of type A
The \(A_\ell\) type Macdonald polynomials \(P_\lambda(z;q,t)\ ,\) see [M1], are certain symmetric homogeneous polynomials in \(\ell+1\) variables of degree \(|\lambda|\) which form an orthogonal system. They can be expressed in terms of \(q\mbox{-}\)ultraspherical polynomials for \(\ell=1\ :\) \[P_{m,n}(x,y;q,t)= \frac{(q;q)_{m-n}}{(t;q)_{m-n}}\,(xy)^{\frac12(m+n)}\,C_{m-n}\biggl(\frac{x+y}{2(xy)^{1/2}};t|q\biggr)\quad(m\ge n\ge0).\] In particular, \[P_{m,0}(e^{i\theta},e^{-i\theta};q,t)=\frac{(q;q)_m}{(t;q)_m}\,C_m(\cos\theta;t|q).\] In the limit for \(q\) to 0 the \(A_\ell\) type Macdonald polynomials are known as Hall-Littlewood polynomials.
Macdonald-Koornwinder polynomials
Macdonald [M2] introduced Macdonald polynomials for all irreducible root systems. They are certain Weyl group invariant trigonometric polynomials forming an orthogonal system and depending on as many parameters (apart from \(q\)) as there are root lengths. Thus the Macdonald polynomials for root system \(BC_\ell\) depend on three parameters. For root system \(BC_1\) this turns down to the two-parameter family of continuous \(q\mbox{-}\)Jacobi polynomials. Koornwinder [Ko] extended the \(BC_\ell\) type Macdonald polynomials to a family depending on five parameters \(a,b,c,d,t\ :\) the Macdonald-Koornwinder polynomials. For \(\ell=1\) they no longer depend on \(t\) and reduce to Askey-Wilson polynomials.
Nonsymmetric Askey-Wilson polynomials
In 1992 Cherednik introduced double affine Hecke algebras (DAHA's) as a natural habitat for nonsymmetric Macdonald polynomials from which the Macdonald polynomials themselves can be obtained by Weyl group symmetrization. In 1999 Sahi extended this approach to Macdonald-Koornwinder polynomials. Its one-variable case led to nonsymmetric Askey-Wilson polynomials in the context of a rather simple DAHA. In the so-called basic representation of this DAHA on the space of Laurent polynomials in one variable a certain element \(Y\) acts on a Laurent polynomial \(f(z)\) as a \(q\mbox{-}\)difference-reflection operator, sending \(f(z)\) to a linear combination of terms \(f(z),f(qz),f(z^{-1}),f(qz^{-1})\) with rational functions in \(z\) as coefficients. It has eigenfunctions \(E_n(z)\) for each integer \(n\ ,\) where \(E_n(z)\) is a linear combination of \(z^{-n},\ldots,z^n\) for \(n>0\ ,\) \(E_{-n}(z)\) is a linear combination of \(z^{-n},\ldots,z^{n-1}\) for \(n>0\ ,\) and \(E_0(z)=1\ .\) The operator \(Y\) has an inverse which is also a \(q\mbox{-}\)difference-reflection operator and the operator \(Y+Y^{-1}\) has two-dimensional eigenspaces spanned by \(E_{\pm n}(z)\ .\) A certain symmetrization operator projects these eigenspaces on one-dimensional spaces of symmetric Laurent polynomials spanned by the Askey-Wilson polynomials \(p_n(\tfrac12(z+z^{-1}))\ .\)
References
- [AW] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs Amer. Math. Soc. 54 (1985), no. 319; MR0783216.
- [GR] G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge University Press, second ed., 2004; MR2128719.
- [KLS] R. Koekoek, P.A. Lesky and R.F. Swarttouw, Hypergeometric orthogonal polynomials and their \(q\)-analogues, Springer-Verlag, 2010; MR2656096.
- [KS] R. Koekoek and R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its \(q\)-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998; online.
- [Ko] T.H. Koornwinder, Askey-Wilson polynomials for root systems of type BC, in: Hypergeometric functions on domains of positivity, Jack polynomials, and applications, D.St.P. Richards (ed.), Contemp. Math. 138, Amer. Math. Soc., 1992, pp. 189-204; MR1199128.
- [M1] I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Second edition, 1994; MR1354144.
