In mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the cl ...
in several variables, introduced by and
I. G. Macdonald (1987, important special cases), that generalize the
Askey–Wilson polynomials. They are the
Macdonald polynomials attached to the non-reduced affine root system of type (''C'', ''C''
''n''), and in particular satisfy (, ) analogues of
Macdonald's conjectures . In addition
Jan Felipe van Diejen
Jan, JaN or JAN may refer to:
Acronyms
* Jackson, Mississippi (Amtrak station), US, Amtrak station code JAN
* Jackson-Evers International Airport, Mississippi, US, IATA code
* Jabhat al-Nusra (JaN), a Syrian militant group
* Japanese Article Numb ...
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 . 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 . The Macdonald-Koornwinder polynomials have also been studied with the aid of
affine Hecke algebras (, , ).
The Macdonald-Koornwinder polynomial in ''n'' variables associated to the partition λ is the unique
Laurent polynomial
In mathematics, a Laurent polynomial (named
after Pierre Alphonse Laurent) in one variable over a field \mathbb is a linear combination of positive and negative powers of the variable with coefficients in \mathbb. Laurent polynomials in ''X'' f ...
invariant under permutation and inversion of variables, with leading monomial ''x''
λ, 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
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^),
with (a;q)_0 = 1.
It is a ''q''-analog of the Pochhammer sym ...
.
Here leading monomial ''x''
λ means that μ≤λ for all terms ''x''
μ with nonzero coefficient, where μ≤λ if and only if μ
1≤λ
1, μ
1+μ
2≤λ
1+λ
2, …, μ
1+…+μ
''n''≤λ
1+…+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.
For some lecture notes on Macdonald-Koornwinder polynomials from a Hecke algebra perspective see for example .
References
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*{{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
Orthogonal polynomials