On Laurent biorthogonal polynomials and Painlevé-type equations

Yue, Xiao-Lu; Chang, Xiang-Ke; Hu, Xing-Biao; Liu, Ya-Jie

doi:10.1090/proc/16037

On Laurent biorthogonal polynomials and Painlevé-type equations
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by Xiao-Lu Yue, Xiang-Ke Chang, Xing-Biao Hu and Ya-Jie Liu PDF

Proc. Amer. Math. Soc. 150 (2022), 4369-4381 Request permission

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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