The monic Laguerre polynomials preserve real-rootedness

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.