Koornwinder polynomials

In mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced by Koornwinder^[1] and I. G. Macdonald,^[2] 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.^[3] 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.^[4] 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.^[5] The Macdonald-Koornwinder polynomials have also been studied with the aid of affine Hecke algebras.^[6]

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

${\displaystyle \prod _{1\leq i<j\leq n}{\frac {(x_{i}x_{j},x_{i}/x_{j},x_{j}/x_{i},1/x_{i}x_{j};q)_{\infty }}{(tx_{i}x_{j},tx_{i}/x_{j},tx_{j}/x_{i},t/x_{i}x_{j};q)_{\infty }}}\prod _{1\leq i\leq n}{\frac {(x_{i}^{2},1/x_{i}^{2};q)_{\infty }}{(ax_{i},a/x_{i},bx_{i},b/x_{i},cx_{i},c/x_{i},dx_{i},d/x_{i};q)_{\infty }}}}$

on the unit torus

${\displaystyle |x_{1}|=|x_{2}|=\cdots |x_{n}|=1}$,

where the parameters satisfy the constraints

${\displaystyle |a|,|b|,|c|,|d|,|q|,|t|<1,}$

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.

Citations

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, MR 1199128, S2CID 14028685
• van Diejen, Jan F. (1996), "Self-dual Koornwinder-Macdonald polynomials", Inventiones Mathematicae, 126 (2): 319–339, arXiv:q-alg/9507033, Bibcode:1996InMat.126..319V, doi:10.1007/s002220050102, MR 1411136, S2CID 17405644
• Sahi, S. (1999), "Nonsymmetric Koornwinder polynomials and duality", Annals of Mathematics, Second Series, 150 (1): 267–282, arXiv:q-alg/9710032, doi:10.2307/121102, JSTOR 121102, MR 1715325, S2CID 8958999
• van Diejen, Jan F. (1995), "Commuting difference operators with polynomial eigenfunctions", Compositio Mathematica, 95: 183–233, arXiv:funct-an/9306002, MR 1313873
• van Diejen, Jan F. (1999), "Properties of some families of hypergeometric orthogonal polynomials in several variables", Trans. Amer. Math. Soc., 351: 233–70, arXiv:q-alg/9604004, doi:10.1090/S0002-9947-99-02000-0, MR 1433128, S2CID 16214156
• Noumi, M. (1995), "Macdonald-Koornwinder polynomials and affine Hecke rings", Various Aspects of Hypergeometric Functions, Surikaisekikenkyusho Kokyuroku (in Japanese), vol. 919, pp. 44–55, MR 1388325
• Macdonald, I. G. (2003), Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge: Cambridge University Press, pp. x+175, ISBN 978-0-521-82472-9, MR 1976581
• 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, MR 2085855