$q$ -Racah Ensemble and $q$-P$\left (E_7^{(1)}/A_{1}^{(1)}\right )$ Discrete Painlevé Equation
$q$ -Racah Ensemble and $q$-P$\left (E_7^{(1)}/A_{1}^{(1)}\right )$ Discrete Painlevé Equation
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$q$ -Racah 系综和 $q$-P$left (E_7^{(1)}/A_{1}^{(1)} ight )$ 离散 Painlevé 方程
DOI：
10.1093/imrn/rnz211
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发表时间：
2019
影响因子：
1
通讯作者：
Knizel, Alisa
中科院分区：
文献类型：
--
作者：
Dzhamay, Anton;Knizel, Alisa
这篇文章的目的是调查故事中关于正交多项式系综和Painlevé方程之间关系的缺失部分。也就是说，我们考虑了-racah多项式系综，并证明了这种情况下的单区间间隔概率可以通过离散P方程的解来表示。我们的方法还给出了这个方程的一个新的Lax对。这对松弛有一个有趣的额外的对合对称结构。
The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlevé equations. Namely, we consider the-Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete-Pequation. Our approach also gives a new Lax pair for this equation. This Lax pair has an interesting additional involutive symmetry structure.