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.