The Szeg\"o curve and Laguerre polynomials with large negative parameters

Abstract

We study the asymptotic zero distribution of the rescaled Laguerre polynomials, $\displaystyle L_n^{(\alpha_n)}(nz)$, with the parameter $\alpha_n$ varying in such a way that $\displaystyle \lim_{n\rightarrow \infty}\alpha_n/n=-1$. The connection with the so-called Szeg\"{o} curve will be showed.