# Koornwinder polynomials

In mathematics, **Macdonald-Koornwinder polynomials** (also called **Koornwinder polynomials**) are a family of orthogonal polynomials in several variables, introduced by Koornwinder (1992) and I. G. Macdonald (1987, important special cases), 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 (Diejen 1996, Sahi 1999) analogues of Macdonald's conjectures (Macdonald 2003, Chapter 5.3). 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 (Diejen 1995). 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 (Diejen 1999). The Macdonald-Koornwinder polynomials have also been studied with the aid of affine Hecke algebras (Noumi 1995, Sahi 1999, Macdonald 2003).

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

on the unit torus

- ,

where the parameters satisfy the constraints

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 μ_{1}≤λ_{1}, μ_{1}+μ_{2}≤λ_{1}+λ_{2}, …, μ_{1}+…+μ_{n}≤λ_{1}+…+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.

For some lecture notes on Macdonald-Koornwinder polynomials from a Hecke algebra perspective see for example (Stokman 2004).

## References

- van Diejen, Jan F. (1995),
*Commuting difference operators with polynomial eigenfunctions*, Compositio Mathematica,**95**, pp. 183–233, arXiv:funct-an/9306002, MR 1313873 - van Diejen, Jan F. (1996),
*Self-dual Koornwinder-Macdonald polynomials*, Invent. Math.,**126**, pp. 319–339, MR 1411136 - van Diejen, Jan F. (1999),
*Properties of some families of hypergeometric orthogonal polynomials in several variables*, Trans. Amer. Math. Soc.,**351**, pp. 233–70, MR 1433128 - Koornwinder, Tom H. (1992),
*Askey-Wilson polynomials for root systems of type BC*, Contemp. Math.,**138**, pp. 189–204, MR 1199128 - Macdonald, I. G. (2003),
*Affine Hecke algebras and orthogonal polynomials*, Cambridge Tracts in Mathematics,**157**, Cambridge: Cambridge University Press, pp. x+175, doi:10.2277/0521824729, ISBN 978-0-521-82472-9, MR 1976581 - Noumi, M. (1995), "Macdonald-Koornwinder polynomials and affine Hecke rings",
*Various Aspects of Hypergeometric Functions*, Surikaisekikenkyusho Kokyuroku (in Japanese),**919**, pp. 44–55, MR 1388325

- Sahi, S. (1999),
*Nonsymmetric Koornwinder polynomials and duality*, Ann. of Math.(2),**150**, pp. 267–282, MR 1715325 - 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 Sci. Publ., pp. 145–207, MR 2085855