The smallest eigenvalue of large Hankel matrices generated by a singularly perturbed Laguerre weight

Abstract

An asymptotic expression of the orthonormal polynomials ${\mathcal{P}}_N(z)$ as $N\to \infty$, associated with the singularly perturbed Laguerre weight $w_{\hat{I}}\pm (x;t)=x^{\hat{I}}\pm {\mathrm{e}}^{-x-\frac{t}{x}},x\in [0,\infty ),\hat{I}\pm >-1,t\ge 0$ is derived. Based on this, we establish the asymptotic behavior of the smallest eigenvalue, $\hat{I}{»}_N$, of the Hankel matrix generated by the weight $w_{\hat{I}}\pm (x;t)$.