Koornwinder polynomials

In mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced by KoornwinderTemplate:Sfn and I. G. Macdonald, that generalize the Askey–Wilson polynomials. They are the Macdonald polynomials attached to the non-reduced affine root system of type (CTemplate:Su, C_n), and in particular satisfy analogues of Macdonald's conjectures.Template:Sfnm 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.Template:Sfn 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.Template:Sfn The Macdonald-Koornwinder polynomials have also been studied with the aid of affine Hecke algebras.Template:Sfnm

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

$\prod _{1\le i<j\le n}\frac{(x_i{x}_j,x_i/x_j,x_j/x_i,1/x_i{x}_j;q{)}_{\mathrm{\infty }}}{(t{x}_i{x}_j,t{x}_i/x_j,t{x}_j/x_i,t/x_i{x}_j;q{)}_{\mathrm{\infty }}}\prod _{1\le i\le n}\frac{(x_i^2,1/x_i^2;q{)}_{\mathrm{\infty }}}{(a{x}_i,a/x_i,b{x}_i,b/x_i,c{x}_i,c/x_i,d{x}_i,d/x_i;q{)}_{\mathrm{\infty }}}$

on the unit torus

$|x_1|=|x_2|=\cdots |x_n|=1$,

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 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.

References

• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation