Koornwinder polynomials

In mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced by Koornwinder and I. G. Macdonald, that generalize the Askey–Wilson polynomials. They are the Macdonald polynomials attached to the non-reduced affine root system of type (C∨n, C_n), and in particular satisfy analogues of Macdonald's conjectures.({{{1}}}, {{{2}}}) In addition Jan Felipe van Diejen showed that the Macdonald polynomials associated to any classical root system can be expressed as limits or special cases of Macdonald-Koornwinder polynomials and found complete sets of concrete commuting difference operators diagonalized by them. Furthermore, there is a large class of interesting families of multivariable orthogonal polynomials associated with classical root systems which are degenerate cases of the Macdonald-Koornwinder polynomials. The Macdonald-Koornwinder polynomials have also been studied with the aid of affine Hecke algebras.({{{1}}}, {{{2}}}) The Macdonald-Koornwinder polynomial in n variables associated to the partition λ is the unique Laurent polynomial invariant under permutation and inversion of variables, with leading monomial x^λ, and orthogonal with respect to the density

[math]\displaystyle{ \prod_{1\le i\lt j\le n} \frac{(x_i x_j,x_i/x_j,x_j/x_i,1/x_ix_j;q)_\infty}{(t x_ix_j,t x_i/x_j,t x_j/x_i,t/x_ix_j;q)_\infty} \prod_{1\le i\le n} \frac{(x_i^2,1/x_i^2;q)_\infty}{(a x_i,a/x_i,b x_i,b/x_i,c x_i,c/x_i,d x_i,d/x_i;q)_\infty} }[/math]

on the unit torus

[math]\displaystyle{ |x_1|=|x_2|=\cdots|x_n|=1 }[/math],

where the parameters satisfy the constraints

[math]\displaystyle{ |a|,|b|,|c|,|d|,|q|,|t|\lt 1, }[/math]

and (x;q)_∞ denotes the infinite q-Pochhammer symbol. Here leading monomial x^λ means that μ≤λ for all terms x^μ with nonzero coefficient, where μ≤λ if and only if μ₁≤λ₁, μ₁+μ₂≤λ₁+λ₂, …, μ₁+…+μ_n≤λ₁+…+λ_n. Under further constraints that q and t are real and that a, b, c, d are real or, if complex, occur in conjugate pairs, the given density is positive.

References

• Koornwinder, Tom H. (1992), "Askey-Wilson polynomials for root systems of type BC", Contemporary Mathematics 138: 189–204, doi:10.1090/conm/138/1199128 
• van Diejen, Jan F. (1996), "Self-dual Koornwinder-Macdonald polynomials", Inventiones Mathematicae 126 (2): 319–339, doi:10.1007/s002220050102, Bibcode: 1996InMat.126..319V 
• Sahi, S. (1999), "Nonsymmetric Koornwinder polynomials and duality", Annals of Mathematics, Second Series 150 (1): 267–282, doi:10.2307/121102 
• van Diejen, Jan F. (1995), "Commuting difference operators with polynomial eigenfunctions", Compositio Mathematica 95: 183–233 
• van Diejen, Jan F. (1999), "Properties of some families of hypergeometric orthogonal polynomials in several variables", Trans. Amer. Math. Soc. 351: 233–70, doi:10.1090/S0002-9947-99-02000-0 
• Noumi, M. (1995), "Macdonald-Koornwinder polynomials and affine Hecke rings" (in Japanese), Various Aspects of Hypergeometric Functions, Surikaisekikenkyusho Kokyuroku, 919, pp. 44–55 
• Macdonald, I. G. (2003), Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, 157, Cambridge: Cambridge University Press, pp. x+175, ISBN 978-0-521-82472-9 
• Stokman, Jasper V. (2004), "Lecture notes on Koornwinder polynomials", Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, Hauppauge, NY: Nova Science Publishers, pp. 145–207