I have the recursion
\begin{align} p_0 &= a_0, \\ p_n &= a_n + \sum_{m = 0}^{n - 1} b_{n - m} p_m, \quad n \ge 1. \end{align}
I have explicit expressions for $\{ a_n \}$ and $\{ b_n \}$, but they are not easy to work with. Is there an explicit solution for $p_n$ or does the existence of an explicit solution depend on the specific form of $\{ a_n \}$ and $\{ b_n \}$?
For what it is worth, the $p_n$ are probabilities that satisfy $0 < p_n < 1$ and $\sum_{n = 0}^\infty p_n = 1$.