The monic Laguerre polynomials preserve real-rootedness

TL;DR

This paper proves that a specific linear operator involving monic Laguerre and associated Laguerre polynomials preserves the property of having all real roots, extending previous results.

Contribution

It establishes that the operator sending $x^n$ to $(-1)^n n! L_n^eta(x)$ preserves real-rootedness for all $eta \\ge 0$, strengthening Fisk's earlier result.

Findings

The operator preserves real-rootedness for associated Laguerre polynomials with nonnegative parameter.

The result generalizes previous work on Laguerre polynomials.

It provides a new tool for analyzing the roots of polynomial sequences.

Abstract

Let $L_n(x)$ and $L_n^\alpha(x)$ be the $n$th Laguerre and associated Laguerre polynomial respectively. Fisk proved that the linear operator sending $x^n$ to $L_n(x)$ preserves real-rootedness. In this note we prove a stronger result; namely, that when $\alpha\ge 0$, the linear operator sending $x^n$ to $(-1)^n n! L_n^\alpha(x)$ preserves real-rootedness.