Chu-Vandermonde Identity/Extended

Theorem
Let $r, s, \alpha, \beta \in \C$ be complex numbers.

Then:
 * $\ds \sum_{k \mathop \in \Z} \dbinom r {\alpha + k} \dbinom s {\beta - k} = \dbinom {r + s} {\alpha + \beta}$

where $\dbinom r {\alpha + k}$ denotes a binomial coefficient.

Proof
From the Chu-Vandermonde Identity, we have:
 * $\ds \sum_{k \mathop \in \Z} \binom r k \binom s {n - k} = \binom {r + s} n$

Let $n = \alpha + \beta$

Let $k = \alpha + k$

Then:


 * $\ds \sum_{k \mathop \in \Z} \binom r {\alpha + k} \binom s {\alpha + \beta - \paren {\alpha + k} } = \binom {r + s} {\alpha + \beta}$


 * $\ds \sum_{k \mathop \in \Z} \binom r {\alpha + k} \binom s {\beta - k } = \binom {r + s} {\alpha + \beta}$