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, Cn), 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.[3] 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.[4] 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 xλ, and orthogonal with respect to the density

∏1≤i<j≤n(xixj,xi/xj,xj/xi,1/xixj;q)∞(txixj,txi/xj,txj/xi,t/xixj;q)∞∏1≤i≤n(xi2,1/xi2;q)∞(axi,a/xi,bxi,b/xi,cxi,c/xi,dxi,d/xi;q)∞

on the unit torus

|x1|=|x2|=⋯|xn|=1,

where the parameters satisfy the constraints

|a|,|b|,|c|,|d|,|q|,|t|<1,

and (x;q)∞ denotes the infinite q-Pochhammer symbol. Here leading monomial xλ means that μ≤λ for all terms xμ with nonzero coefficient, where μ≤λ if and only if μ1≤λ1, μ1+μ2≤λ1+λ2, …, μ1+…+μn≤λ1+…+λn. 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

1. ↑ Koornwinder 1992.
2. ↑ Macdonald 1987, important special cases[full citation needed]
3. ↑ van Diejen 1995.
4. ↑ van Diejen 1999.

## 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

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