The Smallest Eigenvalue of Large Hankel Matrices Generated by a Deformed Laguerre Weight
arXiv:1803.11322 · doi:10.1002/mma.5583
Abstract
We study the asymptotic behavior of the smallest eigenvalue, $λ_{N}$, of the Hankel (or moments) matrix denoted by $\mathcal{H}_{N}=\left(μ_{m+n}\right)_{0\leq m,n\leq N}$, with respect to the weight $w(x)=x^α{\rm e}^{-x^β},~x\in[0,\infty),~α>-1,~β>\frac{1}{2}$. Based on the research by Szegö, Chen, etc., we obtain an asymptotic expression of the orthonormal polynomials $\mathcal{P}_{N}(z)$ as $N\rightarrow\infty$, associated with $w(x)$. Using this, we obtain the specific asymptotic formulas of $λ_{N}$ in this paper. Applying the parallel algorithm discovered by Emmart, Chen and Weems, we get a variety of numerical results of $λ_{N}$ corresponding to our theoretical calculations.
23 pages, 1 figure