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

An asymptotic expression of the orthonormal polynomials ${\mathcal{P}}_N(z)$ as $N\to \mathrm{\infty }$, associated with the singularly perturbed Laguerre weight $w_{\alpha}(x;t)=x^{\alpha}{\rm e}^{-x-\frac{t}{x}},~x\in[0,\infty),~\alpha>-1,~t\geq0$ is derived. Based on this, we establish the asymptotic behavior of the smallest eigenvalue, ${\lambda }_N$, of the Hankel matrix generated by the weight $w_{\alpha }(x;t)$.