I've tried everything but can't quite figure out how to prove this entity using non-induction approach. Anyone have any idea on this entity?
$${\sum_{k}}\binom{l}{m + k}\binom{s + k}{n}(-1)^{k} = (-1)^{l + m}\binom{s - m}{n - l}, l \geq 0; n, m \in Z$$.
Concrete mathematics authors quote:
The next one, (5.24), is a bit more dicult. We can reduce it to Vandermonde's convolution by a sequence of transformations.
Vandermonde's convolution: $${\sum_{k}}\binom{r}{m + k}\binom{s}{n - k} = \binom{r + s}{n + m}$$.
I found this link: Seemingly wrong approach. They applied pascal's rule for $\binom{s + k}{n}$ but $s$ is in fact real !?