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In mathematics, '''Koornwinder polynomials''' (also called '''Macdonald–Koornwinder polynomials''') are a family of [[orthogonal polynomials]] in several variables, introduced by {{harvs|txt|authorlink=Tom H. Koornwinder|last=Koornwinder|year=1992}}, 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=&or;}}, ''C''<sub>''n''</sub>), and in particular satisfy ({{harvnb|Diejen|1996}}, {{harvnb|Sahi|1999}}) analogues of [[Macdonald polynomials|Macdonald's conjectures]] {{harv|Macdonald|2003|loc= 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 Koornwinder polynomials and found complete sets of concrete commuting difference operators diagonalized by them {{harv|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 Koornwinder-Macdonald polynomials {{harv|Diejen|1999}}. The Koornwinder-Macdonald polynomials have also been studied with the aid of [[affine Hecke algebra]]s ({{harvnb|Noumi|1995}}, {{harvnb|Sahi|1999}}, {{harvnb|Macdonald|2003}}).
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The 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''<sup>λ</sup>, and orthogonal with respect to the density
 
:<math> \prod_{1\le i<j\le n} \frac{(x_i x_j,x_i/x_j,x_j/x_i,1/x_ix_j;q)_\infty}{(t x_ix_j,t x_i/x_j,t x_j/x_i,t/x_ix_j;q)_\infty}
\prod_{1\le i\le n} \frac{(x_i^2,1/x_i^2;q)_\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)_\infty}
</math>
 
on the unit torus
 
:<math> |x_1|=|x_2|=\cdots|x_n|=1</math>,
 
where the parameters satisfy the constraints
 
:<math>|a|,|b|,|c|,|d|,|q|,|t|<1, </math>
 
and (''x'';''q'')<sub>∞</sub> denotes the infinite [[q-Pochhammer symbol]].
Here leading monomial ''x''<sup>λ</sup> means that μ≤λ for all terms  ''x''<sup>μ</sup> with nonzero coefficient, where μ≤λ if and only if μ<sub>1</sub>≤λ<sub>1</sub>, μ<sub>1</sub>+μ<sub>2</sub>≤λ<sub>1</sub>+λ<sub>2</sub>, …, μ<sub>1</sub>+…+μ<sub>''n''</sub>≤λ<sub>1</sub>+…+λ<sub>''n''</sub>.
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.
 
For some lecture notes on Koornwinder polynomials from a Hecke algebra perspective see for example  {{harv|Stokman|2004}}.
 
==References==
 
*{{Citation | last1=van Diejen | first1=Jan F. | title=Commuting difference operators with polynomial eigenfunctions | series=Compositio Math. | volume=95 | year=1995 | pages=183–233 | id={{MathSciNet|id=1313873}}  }}
 
*{{Citation | last1=van Diejen | first1=Jan F. | title=Self-dual Koornwinder-Macdonald polynomials | series=Invent. Math. | volume=126 | year=1996 | pages=319–339 | id={{MathSciNet|id=1411136}}  }}
 
*{{Citation | last1=van Diejen | first1=Jan F. | title=Properties of some families of hypergeometric orthogonal polynomials in several variables | series=Trans. Amer. Math. Soc.| volume=351 | year=1999 | pages=233–70 | id={{MathSciNet|id=1433128}}  }}
 
*{{Citation | last1=Koornwinder | first1=Tom H. | title=Askey-Wilson polynomials for root systems of type BC | series=Contemp. Math. | volume=138 | year=1992 | pages=189–204 | id={{MathSciNet|id=1199128}}}}
 
*{{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| doi=10.2277/0521824729 | id={{MathSciNet|id=1976581}}}}
 
*{{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| 
id={{MathSciNet|id=1388325}}  }}
 
*{{Citation | last1=Sahi | first1=S. | title=Nonsymmetric Koornwinder polynomials and duality | series=Ann. of Math.(2)| volume=150 | year=1999 | pages=267–282 | id={{MathSciNet|id=1715325}}  }}
 
*{{Citation | last1=Stokman | first1=Jasper V. | title=Laredo Lectures on Orthogonal Polynomials and Special Functions | publisher=Nova Sci. Publ. | location=Hauppauge, NY | series=Adv. Theory Spec. Funct. Orthogonal Polynomials | id={{MathSciNet | id = 2085855}} | year=2004 | chapter=Lecture notes on Koornwinder polynomials | pages=145–207}}
 
[[Category:Orthogonal polynomials]]

Revision as of 18:40, 28 February 2014

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