Dixon's Identity/General Case

Theorem
For $l, m, n \in \Z_{\ge 0}$:


 * $\displaystyle \sum_{k \mathop \in \Z} \left({-1}\right)^k \dbinom {l + m} {l + k} \dbinom {m + n} {m + k} \dbinom {n + l} {n + k} = \dfrac {\left({l + m + n}\right)!} {l! \, m! \, n!}$

Proof
From Sum over $k$ of $\dbinom {m + r + s} k$ by $\dbinom {n + r - s} {n - k}$ by $\dbinom {r + k} {m + n}$:


 * $\displaystyle \sum_k \binom {m - r + s} k \binom {n + r - s} {n - k} \binom {r + k} {m - n} = \binom r m \binom s n$

Setting $\left({m, n, r, s, k}\right) \gets \left({m + k, l - k, m + n, n + l, j}\right)$:

Unless $j = l$, the sum on $k$ vanishes.