What is the closed form solution to the following partial recurrence relation? $f(k,n) = \sum\limits_{t=0}^{m}f(k-t, n-1),$ where $ m \geq 0$ is some fixed parameter.
The boundary values are $f(k,n)=0$ for all $k < 0, n < 1$ except $f(0,0)=1$.
I noticed that for $m \geq k$ its closed form is a binomial coefficient $f(k,n)=C_{k+n-1}^{n-1}$, $k \geq 0$, $n > 0$. Do you have any idea how to solve this equation for $m