Let $L_n$ be the $n$-th Laguerre polynomial defined by $\quad L_n (x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^n e^{-x}).\quad $ I want to prove that $$ \forall n\in \mathbb N,\forall x\ge 0,\quad \sum_{0\le k\le n}(-1)^kL_k(x)\ge 0. $$ Does anybody know a proof of this inequality or a reference ?

This inequality follows from (6.8) in Positive Jacobi polynomial sums II by Askey and Gasper, which says that $$ \sum_{k=0}^n (-1)^k L_k(x) = \sum_{j=0}^{\lfloor n/2 \rfloor} \frac{2^{-n} (2j)!}{j!^2 (n-2j)!} H_{n-2j}^2\left(\sqrt{x/2}\right). $$ They give a couple of earlier references for this identity, but I haven't looked them up. (So I'm just giving a concrete reference, not saying this was the first proof.)