Asymptotic zero behavior of Laguerre polynomials with negative parameter

Abstract

We consider Laguerre polynomials $L_n^{(\alpha_n)}(nz)$ with varying negative parameters $\alpha_n$, such that the limit $A = -\lim_n \alpha_n/n$ exists and belongs to $(0,1)$. For $A > 1$, it is known that the zeros accumulate along an open contour in the complex plane. For every $A \in (0,1)$, we describe a one-parameter family of possible limit sets of the zeros. Under the condition that the limit $r= - \lim_n \frac{1}{n} \log \dist(\alpha_n, \mathbb Z)$ exists, we show that the zeros accumulate on $\Gamma_r \cup [\beta_1,\beta_2]$ with $\beta_1$ and $\beta_2$ only depending on $A$. For $r \in [0,\infty)$, $\Gamma_r$ is a closed loop encircling the origin, which for $r = +\infty$, reduces to the origin. This shows a great sensitivity of the zeros to $\alpha_n$'s proximity to the integers. We use a Riemann-Hilbert formulation for the Laguerre polynomials, together with the steepest descent method of Deift and Zhou to obtain asymptotics for the polynomials, from which the zero behavior follows.