Generalized Airy polynomials, Hankel determinants and asymptotics

TL;DR

This paper investigates generalized Airy polynomials, deriving their recurrence relations, asymptotic behaviors, and Hankel determinants through ladder operators, Toda equations, and Coulomb fluid methods, extending understanding of their asymptotic properties.

Contribution

It introduces a discrete system for recurrence coefficients of generalized Airy polynomials and establishes their large n asymptotics, including Hankel determinants, using advanced analytical techniques.

Findings

Derived ladder operator equations and compatibility conditions.

Established relations between recurrence coefficients and Hankel determinants.

Obtained large n asymptotic expansions for key quantities.

Abstract

We further study the orthogonal polynomials with respect to the generalized Airy weight based on the work of Clarkson and Jordaan [{\em J. Phys. A: Math. Theor.} {\bf 54} ({2021}) {185202}]. We prove the ladder operator equations and associated compatibility conditions for orthogonal polynomials with respect to a general Laguerre-type weight of the form $w(x)=x^\lambda w_0(x),\;\lambda>-1, x\in\mathbb{R}^+$. By applying them to the generalized Airy polynomials, we are able to derive a discrete system for the recurrence coefficients. Combining with the Toda evolution, we establish the relation between the recurrence coefficients, the sub-leading coefficient of the monic generalized Airy polynomials and the associated Hankel determinant. Using Dyson's Coulomb fluid approach and with the aid of the discrete system for the recurrence coefficients, we obtain the large $n$ asymptotic expansions for the recurrence coefficients and the sub-leading coefficient of the monic generalized Airy polynomials. The large $n$ asymptotic expansion (including the constant term) of the Hankel determinant has been derived by using a recent result in the literature. The long-time asymptotics of these quantities have also been discussed explicitly.