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.

References in corpus (1)

• The Laguerre polynomials preserve real-rootedness