Dixon's Identity/General Case

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


 * $\ds \sum_{k \mathop \in \Z} \paren {-1}^k \dbinom {l + m} {l + k} \dbinom {m + n} {m + k} \dbinom {n + l} {n + k} = \dfrac {\paren {l + m + n}!} {l! \, m! \, n!}$

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


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

Setting $\tuple {m, n, r, s, k} \gets \tuple {m + k, l - k, m + n, n + l, j}$ into the equation above, we obtain:

On the, we have:

Unless $j = l$ and $k = 0$ the sum on $k$ vanishes.

Therefore:
 * $\ds \sum_{k \mathop \in \Z} \paren {-1}^k \dbinom {l + m} {l + k} \dbinom {m + n} {m + k} \dbinom {n + l} {n + k} = \dfrac {\paren {l + m + n}!} {l! \, m! \, n!}$