Equiconvergence for perturbed Jacobi polynomial expansions

Critical edge behavior in the singularly perturbed Pollaczek-Jacobi type unitary ensemble

In this paper, we study the strong asymptotic for the orthogonal polynomials and universality associated with singularly perturbed Pollaczek-Jacobi type weight $$w_{p_J2}(x,t)=e^{-\frac{t}{x(1-x)}}x^α(1-x)^β, $$ where $t \ge 0$, $α>0$, $β>0$ and $x \in [0,1].$ Our main results obtained here include two aspects: { I. Strong asymptotics:} We obtain the strong asymptotic expansions for the monic Pollaczek-Jacobi type orthogonal polynomials in different interval $(0,1)$ and outside of interval $\mathbb{C}\backslash (0,1)$, respectively; Due to the effect of $\frac{t}{x(1-x)}$ for varying $t$, diff…

作为受扰Duffing方程的受限三体问题

The Restricted Three-Body Problem as a Perturbed Duffing Equation

本文通过坐标变换将受限圆形平面三体问题化为受扰Duffing方程，证明在足够大的负雅可比常数下，除有限多个质量比外均产生同宿缠结，且证明更简单直接、自洽。

与受扰动的 $R_I$ 型递推关系相关的谱变换

Spectral transformation associated with a perturbed $R_I$ type recurrence relation

本文研究受扰动 $R_I$ 型递推关系下正交多项式的结构与谱变换，证明传递矩阵法更高效，并分析共递推、共膨胀及零点性质。

Differential, Difference and Asymptotic Relations for Pollaczek-Jacobi Type Orthogonal Polynomials and Their Hankel Determinants

In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek-Jacobi type weight $$ w(x,t):=(1-x^2)^α\mathrm{e}^{-\frac{t}{1-x^{2}}},\qquad x\in[-1,1],\;\;α>0,\;\;t>0. $$ By using the ladder operator approach, we establish the second-order difference equations satisfied by the recurrence coefficient $β_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of the monic orthogonal polynomials, respectively. We show that the logarithmic derivative of $β_n(t)$ can be expressed in terms of a particular Painlevé V transcendent. The large $n$ asymptotic expansi…

混合多重正交多项式的一般 Geronimus 扰动

General Geronimus Perturbations for Mixed Multiple Orthogonal Polynomials

本文研究由一般正则矩阵多项式定义的 Geronimus 变换对混合多重正交多项式的作用，建立 Christoffel 型公式，证明扰动正交性存在等价于 τ-行列式不消没，并以三权重 Jacobi-Piñeiro 多项式为例进行验证。

阶梯线上混合型多重正交多项式的Uvarov扰动

Uvarov Perturbations for Multiple Orthogonal Polynomials of the Mixed Type on the Step-Line

本文在矩阵分析框架下统一了Christoffel、Geronimus和Uvarov变换，研究了阶梯线上混合型多重正交多项式的Uvarov型扰动，给出了连接公式与行列式条件，并以Jacobi-Piñeiro多项式为例说明。