Asymptotic Properties of Some Freud Polynomials

TL;DR

This paper investigates the asymptotic behavior of Freud polynomials, focusing on recurrence coefficients, Hankel determinants, and norms, using advanced analytical methods to deepen understanding of their properties as degree increases.

Contribution

It provides new asymptotic formulas for Freud polynomials' recurrence coefficients, Hankel determinants, and norms, combining ladder operator and Coulomb fluid approaches.

Findings

Asymptotic formulas for recurrence coefficients

Asymptotics of Hankel determinants

Norms of Freud polynomials

Abstract

We study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Freud weights when the degree of the polynomial tends to infinity, including the asymptotics of the recurrence coefficients, the nontrivial leading coefficients of the monic OPs, the associated Hankel determinants and the squares of $L^2$-norm of the monic OPs. These results are derived from the combination of the ladder operator approach, Dyson's Coulomb fluid approach and some recent results in the literature.