Orthogonal polynomials for the singularly perturbed Laguerre weight, Hankel determinants and asymptotics
Chao Min, Xiaoqing Wu
TL;DR
This work analyzes orthogonal polynomials with the singularly perturbed Laguerre weight $w(x;t,\alpha)$ and the associated Hankel determinants $D_n(t)$. It extends the ladder-operator framework to produce a second-order differential equation for $P_n$, a discrete system for the recurrence coefficients $(\alpha_n,\beta_n)$ and auxiliary quantities, and a Toda-type differential-difference system, all linked to $D_n(t)$ and the subleading coefficient $p(n,t,\alpha)$. The authors obtain comprehensive large-$n$ asymptotics for $\alpha_n(t)$, $\beta_n(t)$, $p(n,t)$, $H_n(t)$, $D_n(t)$, and $h_n(t)$ for fixed $t>0$, including explicit $t$-dependent terms with fractional powers, and analyze long-time behavior as $t\to+\infty$; the expansions reveal singular behavior at $t=0$ and connect to Painlevé III$'$ in the sigma form, consistent with RH-based results by Xu, Dai and Zhao. Collectively, these results provide detailed, term-rich asymptotics for a semi-classical Laguerre-type model, with implications for random matrix theory, integrable systems, and related physical models.
Abstract
Based on the work of Chen and Its [{\em J. Approx. Theory} {\bf 162} ({2010}) {270--297}], we further study orthogonal polynomials with respect to the singularly perturbed Laguerre weight $w(x;t,α) = {x^α}{\mathrm e^{- x-\frac{t}{x}}}, \; x\in\mathbb{R}^{+},\;α> -1,\; t\geq 0$. By using the ladder operators and associated compatibility conditions for orthogonal polynomials with general Laguerre-type weights, we derive the second-order differential equation satisfied by the orthogonal polynomials, a system of difference equations and a system of differential-difference equations for the recurrence coefficients. We also investigate the properties of the zeros of the orthogonal polynomials. Using Dyson's Coulomb fluid approach together with the discrete system, we obtain the large $n$ asymptotic expansions of the recurrence coefficients, the sub-leading coefficient of the monic orthogonal polynomials, the Hankel determinant and the normalized constant for fixed $t>0$. It is found that all the asymptotic expansions are singular at $t=0$. We also study the long-time ($t\rightarrow+\infty$) asymptotics of these quantities explicitly for fixed $n\in\mathbb{N}$ from the Toda-type system.