I have a problem, which is probably quite trivial. Consider a recurrence relation of the form
$ C_m = \alpha_m C_{m-1} + \beta_m C_{m-2}, $ where the coefficients $\alpha_m$ and $\beta_m$ are non-commuting. In my problem, the $C_m$'s are functions of certain variables and the coefficients are (slightly complicated) differential operators. Now assuming that $C_{-1} = 0$, I want to obtain a relation of the form $ C_m = P_m(\alpha, \beta)C_0, $ where $P(\alpha, \beta)$ is a polynomial in $\{\alpha_n\}_{n=1}^{m}$ and $\{\beta_n\}_{n=1}^{m}$. For example for $\beta_n = 0$, it's given by $P_m(\alpha,0) = \prod_{n=0}^{m-1}\alpha_{m-n}= \alpha_m\alpha_{m-1}\dots\alpha_1$. If one iterates the first equation, a certain pattern emerges, but it's not clear how to write down a compact expression for $P_m(\alpha,\beta)$.
I guess there are certain standard tricks for these kind of problems, any suggestions?