The monic Laguerre polynomials preserve real-rootedness

Let $L_n(x)$ and $L_n^α(x)$ 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 $α\ge 0$, the linear operator sending $x^n$ to $(-1)^n n! L_n^α(x)$ preserves real-rootedness.

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