Perturbed Hankel Determinants

TL;DR

This paper derives the large n asymptotics of determinants of perturbed Hankel matrices from Jacobi weights, providing explicit formulas and constants for these determinants in the context of smooth weight perturbations.

Contribution

It introduces a method to compute the asymptotics of Hankel determinants with smooth perturbations of Jacobi weights, including the missing constant term.

Findings

Derived explicit large n asymptotics for perturbed Hankel determinants.

Computed the missing constant in the asymptotic expansion.

Extended previous methods to include smooth perturbations of classical weights.

Abstract

In this short note, we compute, for large n the determinant of a class of n x n Hankel matrices, which arise from a smooth perturbation of the Jacobi weight. For this purpose, we employ the same idea used in previous papers, where the unknown determinant, D_n[w_{\alpha,\beta}h] is compared with the known determinant D_n[w_{\alpha,\beta}]. Here w_{\alpha,\beta} is the Jacobi weight and w_{\alpha,\beta}h, where h=h(x),x\in[-1,1] is strictly positive and real analytic, is the smooth perturbation on the Jacobi weight w_{\alpha,\beta}(x):=(1-x)^\alpha (1+x)^\beta. Applying a previously known formula on the distribution function of linear statistics, we compute the large n asymptotics of D_n[w_{\alpha,\beta}h] and supply a missing constant of the expansion.