Orthogonal polynomials for the singularly perturbed Laguerre weight, Hankel determinants and asymptotics

TL;DR

This paper analyzes orthogonal polynomials with a singularly perturbed Laguerre weight, deriving differential equations, asymptotic expansions, and studying zero properties, with implications for Hankel determinants and long-time behavior.

Contribution

It introduces new differential equations and asymptotic formulas for orthogonal polynomials with a perturbed Laguerre weight, extending prior work and exploring zero distributions.

Findings

Derived second-order differential equations for the polynomials

Obtained large n asymptotic expansions of recurrence coefficients

Analyzed long-time asymptotics for fixed n

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 $w(x;t,\alpha) = {x^\alpha}{\mathrm e^{- x-\frac{t}{x}}}, \; x\in\mathbb{R}^{+},\;\alpha > -1,\; t\geq 0$. 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.