Difference equations and symmetric polynomials defined by their zeros
Difference equations and symmetric polynomials defined by their zeros
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差分方程和由零点定义的对称多项式
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
S. Sahi
中科院分区：
文献类型：
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作者：
F. Knop;S. Sahi
在本文中，我们开始系统地分析一类对称多项式，在充分的一般性，已介绍了[Sa]。这些函数的主要特征是它们是由消失条件定义的，并且它们是非齐次的。它们依赖于几个参数，但我们主要研究的是由一个参数r索引的某个子族。作为一种特殊情况，我们得到了Biedenharn和Louck [BL]在r = 1时发现的阶乘Schur函数.我们的主要结果是，对于一般的r，这些功能的特征值的差分算子，这是不同的类似物的关口-Debiard微分算子。因此，所研究的函数是杰克多项式的非齐次变体。更准确地说，设Λ是长度为n的分区的集合，即，整数序列（λi），其中λ1 ≥ . . .≥ λn ≥ 0。程度|λ|分划λ是其各部分之和。选择一个向量，它必须满足一个温和的条件。则对于每个λ ∈ Λ，存在（直到常数）唯一的次数至多为d的对称多项式Pλ，其满足以下消失条件：
In this paper, we are starting a systematic analysis of a class of symmetric polynomials which, in full generality, has been introduced in [Sa]. The main features of these functions are that they are defined by vanishing conditions and that they are non-homogeneous. They depend on several parameters but we are studying mainly a certain subfamily which is indexed by one parameter r. As a special case, we obtain for r = 1 the factorial Schur functions discovered by Biedenharn and Louck [BL]. Our main result is that for general r these functions are eigenvalues of difference operators, which are difference analogues of the Sekiguchi-Debiard differential operators. Thus the functions under investigation are non-homogeneous variants of Jack polynomials. More precisely, let Λ be the set of partitions of length n, i.e., sequences of integers (λi) with λ1 ≥ . . . ≥ λn ≥ 0. The degree |λ| of a partition λ is the sum of its parts. Choose a vector ̺ ∈ C which has to satisfy a mild condition. Then for every λ ∈ Λ there is (up to a constant) a unique symmetric polynomial Pλ of degree at most d which satisfies the following vanishing condition: