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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?

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    I think the "standard tricks" are for the commuting case.2012-03-22

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I don't think there is an easy notation: it might be easier to program.

Let $N_{\le m}$ be the set of positive integers up to $m$, i.e. $\{1,2,3,\ldots,m\}$. Let $T_m$ comprise all $S \subset N_{\le m-1}$ where you have $s \in S \implies s+1 \not \in S$; $T_m$ is a subset of the powerset of $N_{\le m-1}$. Let $S_{+1}$ be such that $s \in S \iff s+1 \in S_{+1}$. You could define $T_m$ as comprising all $S \subset N_{\le m-1}$ where you have $S \cap S_{+1} = \emptyset.$

Then $P_m(\alpha,\beta) = \sum_{S \in T_m} \left(\prod_{r \in N_{\le m} \backslash (S \cup S_{+1})} \alpha_r \right) \left(\prod_{s \in S} \beta_s \right)$

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    @Heidar: I can't help on the first point. On the second, yo$u$ would have to "decommute" the products so the subscripts were decreasing from left to right.2012-03-26