# 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?