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I am searching a way to solve a recursive definition for a sequence in the following form:

$a_n = c_{n-1}\cdot a_{n-1} + b_{n-1}$

So I'd like to have an explicit form of $a_n$ depending only on $c_i$ and $b_i$, but not on $a_n$. I know how to solve linear recursions, but the coefficients are not constant here. Or are there no well-known solutions for this case?

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As it is, the problem seems a bit too general (any sequence satisfies many such recursions). So while you can express $a_n$ is terms of the $b_i$ and $c_i$, it's not a simple expression in general. Indeed if you set $C_n = \prod_{i = 0}^{n} c_i$, then you can express $a_n$ as

$a_{n+1} = C_n \, \left(a_0 + \sum_{k = 0}^n \frac{b_k}{C_k}\right)$

But unless you have specific sequences $(b_n)_{n \ge 0}$ and $(c_n)_{n \ge 0}$, I doubt you can do much better than that.

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    Sorry, you acutally answered my question completly. I had different expectations and didn't want to see that you are right... ;)2011-11-14