Koornwinder polynomials
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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.[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 xλ, 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 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.