# Recurrence with cases

$\def\left{\operatorname{left}}\def\right{\operatorname{right}}$I have the following recurrence with cases: $$p(l, r, s) = 0.5 \cdot \left(l, r, s) + 0.5 \cdot \right(l, r, s)$$ where: $$\left(l, r, s) = \begin{cases} p(l - s, r, 2s) & s \le l\\ 1 & l < s \le l + w\\ 0 & l + w < s\end{cases}$$

$$\right(l, r, s) = \begin{cases} p(l, r - s, 2 s) & s \le r\\ 1 & r < s \le r + w\\ 0 & r + w < s\end{cases}$$

Is there a way to solve this recurrence analytically?

I'm interested in finding a solution for parameters such that $l = r = k$, $s = 1$ and $w = f\cdot k$, where $f \in (0, 1)$.

If not, is there a way to solve this analytically for a fixed $k$?

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What did you mean by $f \in \langle 0, 1 \rangle$? I replaced it with $f \in (0,1)$, was that right? – martini Dec 10 '12 at 13:35
I meant that w is some fraction of k, that is w is in the interval (0, 1). Thanks for fixing it. – axel22 Dec 10 '12 at 14:17
Please use $...$ for mathematical text. – martini Dec 10 '12 at 14:20
Thanks for letting me know! – axel22 Dec 10 '12 at 14:24