The Smallest Eigenvalue of Large Hankel Matrices Generated by a Deformed Laguerre Weight

TL;DR

This paper analyzes the asymptotic behavior of the smallest eigenvalue of large Hankel matrices generated by a deformed Laguerre weight, providing explicit formulas and numerical validation.

Contribution

It derives asymptotic formulas for the smallest eigenvalue of Hankel matrices with a deformed Laguerre weight, extending previous work and validating results numerically.

Findings

Asymptotic expression for the smallest eigenvalue $\

Numerical results confirming theoretical asymptotics

Explicit formulas for orthonormal polynomials associated with the weight

Abstract

We study the asymptotic behavior of the smallest eigenvalue, $\lambda_{N}$, of the Hankel (or moments) matrix denoted by $\mathcal{H}_{N}=\left(\mu_{m+n}\right)_{0\leq m,n\leq N}$, with respect to the weight $w(x)=x^{\alpha}{\rm e}^{-x^{\beta}},~x\in[0,\infty),~\alpha>-1,~\beta>\frac{1}{2}$. Based on the research by Szeg\"{o}, Chen, etc., we obtain an asymptotic expression of the orthonormal polynomials $\mathcal{P}_{N}(z)$ as $N\rightarrow\infty$, associated with $w(x)$. Using this, we obtain the specific asymptotic formulas of $\lambda_{N}$ in this paper. Applying the parallel algorithm discovered by Emmart, Chen and Weems, we get a variety of numerical results of $\lambda_{N}$ corresponding to our theoretical calculations.