Sum over k of r Choose k by s+k Choose n by -1^r-k/Corollary

Corollary to Sum over $k$ of $\dbinom r k \dbinom {s + k} n \paren {-1}^{r - k}$

Let $r \in \Z_{\ge 0}, n \in \Z$.

Then:

$\ds \sum_k \binom r k \binom k n \paren {-1}^{r - k} = \delta_{n r}$

where $\delta_{n r}$ is the Kronecker delta.

$\ds \sum_k \binom r k \binom {s + k} n \paren {-1}^{r - k} = \binom s {n - r}$

which holds for $s \in \R, r \in \Z_{\ge 0}, n \in \Z$.

Setting $s = 0$:

$\ds \sum_k \binom r k \binom k n \paren {-1}^{r - k} = \binom 0 {n - r}$

We have by definition of binomial coefficient that:

$\dbinom 0 0 = 1$

and:

$\forall n \in \Z_{\ne 0}: \dbinom 0 n = 0$

$\dbinom 0 {n - r} = \sqbrk {n = r}$

The result follows by definition of the Kronecker delta.

$\blacksquare$