probability – Ways of solving the following recurrence relation system.

Consider the following system of linear recurrence relations.

$$begin{aligned} p_n &= a cdot p_{n-1} – c_{n-1}\ c_{n-1} &= p_{n-1} – b cdot p_{n-2} + c_{n-2}end{aligned}$$

with $p_0 = 1$ and $c_1 = 1$.

I’ve tried to represent $p_n$ as a finite linear combination of $p_{k}, k < n$, but this doesn’t work for me. Maybe there is any chance to represent the final solution?

Any ideas?