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In mathematics, Koornwinder polynomials are a family of orthogonal polynomials in several variables, named for their discoverer Tom H. Koornwinder, that generalize the Askey-Wilson polynomials. They can also be viewed as Macdonald polynomials attached to the non-reduced root system of type BC, and in particular satisfy analogues of Macdonald's "conjectures". In addition, the Macdonald polynomials associated to any classical root system can be expressed as limits or special cases of Koornwinder polynomials.

The Koorwinder polynomial in n variables associated to the partition λ is the unique Laurent polynomial invariant under permutation and inversion variables, with leading monomial x^λ, and orthogonal with respect to the density

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

on the unit torus

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

where the parameters satisfy the constraint

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

and $(x;q)_{\infty }$ denotes the infinite q-Pochhammer symbol.

References

• Koornwinder, Tom H. (1992), Askey-Wilson polynomials for root systems of type BC, Contemp. Math., vol. 138, pp. 189–204

• Koornwinder, Tom H. (1974), "Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. I", Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math., 36: 48–58, MR0340673

• 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, MR2085855