Koornwinder polynomials

In mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced by Koornwinder{{sfn|Koornwinder|1992}} and I. G. Macdonald,Macdonald 1987, important special cases{{Full citation needed|date=October 2023}} that generalize the Askey–Wilson polynomials. They are the Macdonald polynomials attached to the non-reduced affine root system of type (C{{su|b=n|p=∨}}, C_n), and in particular satisfy analogues of Macdonald's conjectures.{{sfnm|1a1=van Diejen|1y=1996|2a1=Sahi|2y=1999|3a1=Macdonald|3y=2003|3loc=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.{{sfn|van 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.{{sfn|van Diejen|1999}} The Macdonald-Koornwinder polynomials have also been studied with the aid of affine Hecke algebras.{{sfnm|1a1=Noumi|1y=1995|2a1=Sahi|2y=1999|3a1=Macdonald|3y=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

: $\prod_{1\le i$

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.

Citations

References

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{{Citation | last1=van Diejen | first1=Jan F. | title=Self-dual Koornwinder-Macdonald polynomials | journal=Inventiones Mathematicae | volume=126 | year=1996 | issue=2 | pages=319–339 | doi=10.1007/s002220050102 | arxiv=q-alg/9507033 | bibcode=1996InMat.126..319V |mr=1411136| s2cid=17405644}}
{{Citation | last1=Sahi | first1=S. | title=Nonsymmetric Koornwinder polynomials and duality | journal=Annals of Mathematics |series=Second Series| volume=150 | year=1999 | issue=1 | pages=267–282 | doi=10.2307/121102 | jstor=121102 | arxiv=q-alg/9710032 |mr=1715325 | s2cid=8958999}}
{{Citation | last1=van Diejen | first1=Jan F. | title=Commuting difference operators with polynomial eigenfunctions | journal=Compositio Mathematica | volume=95 | year=1995 | pages=183–233 |mr=1313873 | arxiv=funct-an/9306002}}
{{Citation | last1=van Diejen | first1=Jan F. | title=Properties of some families of hypergeometric orthogonal polynomials in several variables | journal=Trans. Amer. Math. Soc.| volume=351 | year=1999 | pages=233–70 | doi=10.1090/S0002-9947-99-02000-0 |mr=1433128 | s2cid=16214156 | doi-access=free| arxiv=q-alg/9604004 }}
{{Citation | last=Noumi | first=M. | chapter = Macdonald-Koornwinder polynomials and affine Hecke rings| title=Various Aspects of Hypergeometric Functions | language=Japanese| series=Surikaisekikenkyusho Kokyuroku | volume=919 | year=1995 | pages=44–55|mr=1388325}}
{{Citation | last=Macdonald | first=I. G. | title = Affine Hecke algebras and orthogonal polynomials | location=Cambridge | series= Cambridge Tracts in Mathematics | volume=157 | publisher=Cambridge University Press | year=2003 | pages=x+175 | isbn=978-0-521-82472-9|mr=1976581}}
{{Citation | last1=Stokman | first1=Jasper V. | title=Laredo Lectures on Orthogonal Polynomials and Special Functions | publisher=Nova Science Publishers | location=Hauppauge, NY | series=Adv. Theory Spec. Funct. Orthogonal Polynomials |mr=2085855 | year=2004 | chapter=Lecture notes on Koornwinder polynomials | pages=145–207}}

Category:Orthogonal polynomials