periodicity of function

If $f(x+1) + f(x-1) = \sqrt3f(x)$, then what is the period of $f(x)$?

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And what have you tried? –  Ｊ. Ｍ. May 7 '11 at 19:09
Hint: Mathematica says that your $f(x)$ are of the form $f(x)=e^{-\frac{1}{6} i \pi x} \left(a+b e^{\frac{i \pi x}{3}}\right)$ for some real $a,b$ –  Listing May 7 '11 at 19:53
@user3123 : Actually, I think Mathematica's answer is not exhaustive. You can take $a$ and $b$ to be any $1$-periodical function. –  Joel Cohen May 7 '11 at 20:45
ok I did not check that –  Listing May 7 '11 at 22:44
You can start solving the equation $(E_{\lambda})$ (where $\lambda$ is a complex number) :
$f(x+1) = \lambda \, f(x)$
If $\lambda_1$ and $\lambda_2$ are the roots of $X^2 - \sqrt{3} X + 1$, you can check solutions to $(E_{\lambda_1})$ and $(E_{\lambda_2})$ are solution to your equation. Conversely solution of your equation are linear combinations of the previous ones.