Sum over k of r-tk Choose k by s-t(n-k) Choose n-k by r over r-tk/Proof 1/Lemma

Theorem
Let this hold for $\tuple {r, s, t, n}$:
 * $\displaystyle \sum_{k \mathop \ge 0} \binom {r - t k} k \binom {s - t \paren {n - k} } {n - k} \frac r {r - t k} = \binom {r + s - t n} n$

and also for $\tuple {r, s - t, t, n - 1}$.

Then it also holds for $\tuple {r, s + 1, t, n}$.

Proof
Evaluating the equation for $\tuple {r, s - t, t, n - 1}$:

Adding the equation in $\tuple {r, s, t, n}$:

Hence the equation holds for $\tuple {r, s + 1, t, n}$