Ruin probabilities as recurrence sequences in a discrete-time risk process

We apply the theory of linear recurrence sequences to find an expression for the ultimate ruin probability in a discrete-time risk process. We assume the claims follow an arbitrary distribution with support $\{0,1,\dots ,m\}$, for some integer $m\ge 2$. The method requires to find the zeroes of an $m$ degree polynomial and to solve a system of $m$ linear equations. An approximation is derived from the exact ruin formula and several numerical results and plots are provided as examples.