Given a "quadratic" recurrence form: $$A = (a_{n+1}-B)(a_n-C)$$ How do I determine the general solution? I've been trying for about an hour now. If I could have a small hint that would be great. Thanks.
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Is there a reason to expect it to have a "closed form" solution? – flawr Oct 15 '17 at 10:34
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First, solve for $a_{n+1}$ as an expresson in $a_n$. This is a linear fractional (Möbius) transformation. See the Wikipedia article Möbius transformation As such, there are either two fixed points, or else one fixed point. These two possibilities lead to simple recurrence relations for $a_n$. In the first case, you transform the fixed points to $0$ and $\infty$, and in the second case, to $\infty$. It is easier to see this in reverse. Suppose we define by recursion $\;a_{n+1}:=f(a_n)\;$ for some function $f(x).\;$ In the first case, we let $f(x):=cx.\;$ Then $a_n$ forms a geometric progression. In the second case, we let $f(x):=c+x.\;$ Then $a_n$ forms an arithmetic progression.
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Can you expand a little on that last line? After I have the fixed points how do I get recurrence relations for $a_n$. – mtheorylord Oct 15 '17 at 20:28