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 $r= - \lim_n \frac{1}{n} \log \dist(α_n, \mathbb Z)$ 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.

arXiv comment28 pages, 10 figures