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

TL;DR

This paper investigates the asymptotic zero distribution of rescaled Laguerre polynomials with parameters approaching a negative linear ratio, revealing a connection to the Szeg"o curve.

Contribution

It provides new insights into the zero distribution of Laguerre polynomials with large negative parameters and links it to the Szeg"o curve.

Findings

Zero distribution converges to the Szeg"o curve

Asymptotic behavior characterized for large negative parameters

Connection established between Laguerre polynomials and Szeg"o curve

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.