标题： 正交多项式：从Heun方程到Painlevé方程 显示英文标题

标题： Orthogonal polynomials: from Heun equations to Painlevé equations

摘要： 在本文中，我们 {\color{black}研究四种与奇异摄动高斯权函数$w_{\rm SPG}(x)$相关的正交多项式，变形弗雷德权函数$w_{\rm DF}(x)$，跳跃高斯权函数$w_{\rm JG}(x)$和雅可比型权函数$w_{\rm {\color{black}JC}}(x)$。 这些正交多项式满足的二阶线性微分方程以及相关的赫努方程被提出。 利用 [J. Dereziński, A. Ishkhanyan, A. Latosiński, SIGMA 17 (2021), 056] 中的等单变分变形方法，我们将这些赫努方程转化为帕莱夫方程。 有趣的是，通过本工作得到的帕莱夫方程与其它作者研究的相关三项递推系数或辅助量所满足的结果相同。 此外，我们讨论由第一个权函数生成的Hankel行列式的渐近行为，即$w_{\rm SPG}(x)$，在大$s$和小$s$的适当双标度下，其中恢复了Dyson常数。} 显示英文摘要

摘要： In this paper, we {\color{black}study four kinds of polynomials orthogonal with the singularly perturbed Gaussian weight $w_{\rm SPG}(x)$, the deformed Freud weight $w_{\rm DF}(x)$, the jumpy Gaussian weight $w_{\rm JG}(x)$, and the Jacobi-type weight $w_{\rm {\color{black}JC}}(x)$. The second order linear differential equations satisfied by these orthogonal polynomials and the associated Heun equations are presented. Utilizing the method of isomonodromic deformations from [J. Derezi\'{n}ski, A. Ishkhanyan, A. Latosi\'{n}ski, SIGMA 17 (2021), 056], we transform these Heun equations into Painlev\'{e} equations. It is interesting that the Painlev\'{e} equations obtained by the way in this work are same as the results satisfied by the related three term recurrence coefficients or the auxiliaries studied by other authors. In addition, we discuss the asymptotic behaviors of the Hankel determinant generated by the first weight, $w_{\rm SPG}(x)$, under a suitable double scalings for large $s$ and small $s$, where the Dyson's constant is recovered.}