A sum of Laguerre polynomials

Question

I'm looking to find a closed-form expression for the sum $$S = \sum_{n=0}^N e^{-x/2} L_n^{0}(x),$$ where $L_n^{0}$ is the $n$th Laguerre polynomial. Using the formula $$L_n^{\alpha}(x) = \sum_{m=0}^n (-1)^m \binom{n+\alpha}{n-m} \frac{x^m}{m!},$$ this is equivalent to finding a closed form for $$\sum_{n=m}^N \binom{n}{m}.$$ Any help would be appreciated. An asymptotic form would also be useful for the quantity $\frac{S}{N}$.

Answer 1

From the identity $\sum_{n=m}^N \binom{n}{m} = \binom{N+1}{m+1}$ we have that

$$S = e^{-x/2} \sum_{m=0}^N (-1)^m \binom{N+1}{m+1} \frac{x^m}{m!} = e^{-x/2}L_N^{1}(x)$$