TITLE: Is the sum of the first N Laguerre polynomials (with alternating signs) always positive?
QUESTION [10 upvotes]: I have noticed that the following simple sum of Laguerre polynomials (weighted with alternating signs) seems to be positive for any $N$ when $x>0$:
$$\sum_{k=0}^{N}\;(-1)^{k}\;L_{k}(x)$$
More precisely, when $N$ is even, the sum is positive $\forall\;x$ and when $N$  is uneven, the sum has only one zero in $x=0$.
I have checked this numerically up to $N=10{}^{2}$.
I cannot find any reference to this sum in the literature. Are my observations correct?   If this is a known result, could someone point me in the right direction to find a reference or some documentation for this?

REPLY [2 votes]: It is proved in https://arxiv.org/abs/2104.14996 that the Wigner function of the equally weighted mixture of the first $M+1$ Fock states is nonnegative, for all $M>=0$.
This proves that the sum in my question is indeed nonnegative because the Wigner function of this equally weighted mixture has the form (in polar coordinates): $\frac{1}{\left(M+1\right)}\sum_{n=0}^{M}\frac{\left(-1\right)^{n}}{\pi}\:L_{n}\left(2r^{2}\right)\:e^{-r^{2}}$.