Sum over k of r+tk choose k by s-tk choose n-k

Theorem
Let $n \in \Z_{\ge 0}$ be a non-negative integer.

Then:


 * $\displaystyle \sum_k \dbinom {r + t k} k \dbinom {s - t k} {n - k} = \sum_{k \mathop \ge 0} \dbinom {r + s - k} {n - k} t^k$

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

Proof
Let $f \left({r, s, t, n}\right)$ be the function defined as:
 * $\displaystyle f \left({r, s, t, n}\right) := \sum_k \dbinom {r + t k} k \dbinom {s - t k} {n - k}$

We have: