How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say $f(x) = f_0(x)$ for $x \in [0, \sqrt{2})$ is a given function.
My attempt:
Let's write characteristic function $L(\lambda) = 1 - e^{-\lambda} - e^{-\lambda\sqrt{2}}$. We need to solve the equation $L(\lambda) = 0$. There should be a root $\lambda_0 > 0$ (because $L$ is monotonously increasing, $L(0) = -1$ and $\lim_{\lambda \to +\infty} L(\lambda) = 1$), but my first question there is: how many other roots it has? Is it true that it has infinite number of (complex) roots? why?..