Koornwinder polynomials

In mathematics, '''Macdonald-Koornwinder polynomials''' (also called '''Koornwinder polynomials''') are a family of [[orthogonal polynomials]] in several variables, introduced by [[Tom H. Koornwinder|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 polynomials|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 algebra]]s.{{sfnm|1a1=Noumi|1y=1995|2a1=Sahi|2y=1999|3a1=Macdonald|3y=2003}}

In mathematics, '''Macdonald-Koornwinder polynomials''' (also called '''Koornwinder polynomials''') are a family of [[orthogonal polynomials]] in several variables, introduced by [[Tom H. Koornwinder|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 polynomials|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 algebra]]s.{{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

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

and (''x'';''q'')_∞ denotes the infinite [[q-Pochhammer symbol]].

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''.

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.

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.