Okounkov's BC-type interpolation Macdonald polynomials and their q=1 limit
Okounkov's BC-type interpolation Macdonald polynomials and their q=1 limit
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Okounkov 的 BC 型插值 Macdonald 多项式及其 q=1 极限
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
T. Koornwinder
中科院分区：
文献类型：
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作者：
T. Koornwinder
本文调查了与 $A$ 型和 $BC$ 型根系统相关的八类多项式：Jack、Jacobi、Macdonald 和 Koornwinder 多项式以及插值（或移位）Jack 和 Macdonald 多项式及其 $BC$ 型扩展。其中，$BC$ 型插值 Jack 多项式迄今为止可能尚未被观测到。重点放在（大多数）这些多项式的组合公式和二项式公式上。从这些公式中导出的可能的新结果是 Koornwinder 到 Macdonald 多项式的极限、Koornwinder 多项式在两个变量中的显式公式，以及 $BC$ 型 Jacobi 多项式的展开系数以 Jack 多项式表示的组合表达式，该表达式与 Macdonald 的组合表达式不同。对于双变量情况下的最后这些系数，现在以完全不同的方式获得 Koornwinder & Sprinkhuizen (1978) 中的显式表达式。
This paper surveys eight classes of polynomials associated with $A$-type and $BC$-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and interpolation (or shifted) Jack and Macdonald polynomials and their $BC$-type extensions. Among these the $BC$-type interpolation Jack polynomials were probably unobserved until now. Much emphasis is put on combinatorial formulas and binomial formulas for (most of) these polynomials. Possibly new results derived from these formulas are a limit from Koornwinder to Macdonald polynomials, an explicit formula for Koornwinder polynomials in two variables, and a combinatorial expression for the coefficients of the expansion of $BC$-type Jacobi polynomials in terms of Jack polynomials which is different from Macdonald's combinatorial expression. For these last coefficients in the two-variable case the explicit expression in Koornwinder & Sprinkhuizen (1978) is now obtained in a quite different way.