Abstract:In this paper, we investigate Laurent biorthogonal polynomials with a weight function of three parameters, i.e. $z^\alpha e^{-t_1z-\frac {t_2}{z}}, z\in (0,+\infty )$, $(t_1>0,\ t_2>0,\ \alpha \in \mathbb {R})$. First, the structure relation of the Laurent biorthogonal polynomials is found with the aid of biorthogonality. Then we derive an alternate discrete Painlevé II by considering the compatibility condition of the three-term recurrence relation and the structure relation. In addition, we make use of the relativistic Toda chains and nonlinear difference system to obtain two continuous Painlevé-type differential equations.

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中文翻译：

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关于 Laurent 双正交多项式和 Painlevé 型方程

摘要：本文研究了具有三个参数的权函数的 Laurent 双正交多项式，即 $z^\alpha e^{-t_1z-\frac {t_2}{z}}, z\in (0,+\infty )$, $(t_1>0,\ t_2>0,\ \alpha \in \mathbb {R})$。首先，借助双正交性找到了Laurent双正交多项式的结构关系。然后我们通过考虑三项递推关系和结构关系的相容条件推导出一个交替离散Painlevé II。此外，我们利用相对论 Toda 链和非线性差分系统得到两个连续的 Painlevé 型微分方程。

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