There's a problem I've been working on for awhile that involves some hefty functional equations. For example, I may have something along the lines of
$ f(x)f(x) =x+1+f(x+1) $
I've tried several different methods of attack (the farthest I ever got was probably with a power series which didn't yield a recurrence relation) but it never amounts to much. As if that wasn't bad enough, I don't actually know any values of $f(x)$, other than that $\lim_{x\to\infty}f(x)=\infty$. The thing is, I don't actually care about $f(x)$, I only want to know $f(0)$ (analytically) but I can never seem to get two equations for a given point.
I was wondering if anyone had any ideas on how to solve this, or even just some insight into whether or not it can be solved. Thanks!
Edit: Additional facts
- It can be required that $1
- $f(x)$ is strictly increasing
- $f(x)$ is non-negative