Consider a function $f(k,n,p)$ that satisfies the following recurrence:
$$ f(k,n,p) = p\cdot f(k-1, n-1, p) + (1-p) \cdot f(k, n-1, p)\,. $$
This is of course the probability of $k$ successes from $n$ trials, easily satisfied by: $$f(k,n,p) = {n \choose k}p^k(1-p)^{n-k}\,,$$ but I was wondering:
Are there other examples of functions that satisfy this recurrence relation, for $k,n \in \mathbb{N}$ and $p \in \mathbb{R^+}$?