论文标题
PainlevéIV，Chazy II和渐近学，用于半古典Laguerre多项式的复发系数及其Hankel决定因素
Painlevé IV, Chazy II, and Asymptotics for Recurrence Coefficients of Semi-classical Laguerre Polynomials and Their Hankel Determinants
论文作者
论文摘要
本文根据Boelen和van Assche \ Cite {Boolen}，Filipuk等人研究了基于先前的工作的一元半古典laguerre多项式。 \ cite {filipuk}，克拉克森和乔丹\ cite {clarkson}。 Filipuk，Van Assche和Zhang证明了对角线复发系数$α_n（T）$满足第四个Parelevé方程。在本文中，我们表明，非对角线复发系数$β_n（t）$满足Chazy II系统的第一个成员。我们还证明，一元半古典laguerre多项式的子领先系数可以满足连续和离散的jimbo-miwa-okamoto $σ$ - painlevéIV的形式。通过使用Dyson的库仑流体方法以及$α_n（t）$和$β_N（T）$的离散系统，我们获得了复发系数和子领导系数的大$ N $渐近扩展。 Associate Hankel决定因素（包括恒定项）的大$ n $渐近学是根据其积分代表来得出的。
This paper studies the monic semi-classical Laguerre polynomials based on previous work by Boelen and Van Assche \cite{Boelen}, Filipuk et al. \cite{Filipuk} and Clarkson and Jordaan \cite{Clarkson}. Filipuk, Van Assche and Zhang proved that the diagonal recurrence coefficient $α_n(t)$ satisfies the fourth Painlevé equation. In this paper we show that the off-diagonal recurrence coefficient $β_n(t)$ fulfills the first member of Chazy II system. We also prove that the sub-leading coefficient of the monic semi-classical Laguerre polynomials satisfies both the continuous and discrete Jimbo-Miwa-Okamoto $σ$-form of Painlevé IV. By using Dyson's Coulomb fluid approach together with the discrete system for $α_n(t)$ and $β_n(t)$, we obtain the large $n$ asymptotic expansions of the recurrence coefficients and the sub-leading coefficient. The large $n$ asymptotics of the associate Hankel determinant (including the constant term) is derived from its integral representation in terms of the sub-leading coefficient.
