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

Abstract

We study the asymptotic behavior of the smallest eigenvalue, ${\lambda }_N$, of the Hankel (or moments) matrix denoted by ${\mathcal{H}}_N={\left({\mu }_{m+n}\right)}_{0\le m,n\le N}$, with respect to the weight $w(x)=x^{\alpha }{\mathrm{e}}^{-x^{\beta }},x\in [0,\infty ),\alpha >-1,\beta >\frac{1}{2}$. Based on the research by Szegö, Chen, etc., we obtain an asymptotic expression of the orthonormal polynomials ${\mathcal{P}}_N(z)$ as $N\to \infty$, 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.

23 pages, 1 figure