For fixed $l\in \mathbb Z_+$ and $\alpha\in \mathbb R$, define a function $$p_l(\alpha) = \sum_{k=0}^\infty \binom \alpha k \binom{-\alpha}{k+l} e^{-2k}.$$ Due to the exponential suppression $e^{-2k}$, the summation seems to be convergent, which is checked numerically.
By numerical investigation, I found out that $p_l(\alpha)$ has a singularity where $\alpha\in \mathbb Z_{>0}$. The following is the overall graph of $p_{12}(\alpha)$ after fixing $l=12$:
It appears as a smooth function, but when we zoom in near the integers, I found small singularity as marked in green. In all figures, $\alpha$ is zoomed near $-1,0,1,2,3,4,5,6,7$. All horizontal axes are $\alpha$ and all vertical axes are $p_{12}(\alpha)$.
Although the amount of singularity (variation of $p_{12}(\alpha)$) is small, I believe this is larger than the precision that I use. Is this singularity real, or is it just a numerical artifact?
Using the Gaussian hypergeometric function $$p_l(\alpha) = \sum_{k=0}^\infty \binom \alpha k \binom{-\alpha}{k+l} e^{-2k}=\binom{-\alpha }{l} \, _2F_1\left(-\alpha ,l+\alpha ;l+1;\frac{1}{e^2}\right)$$ which has discontinuities at negative integer values of $l$.
Have a look here.