Given: $X$, $Y$ iid random variables, $\mathbb E(X) = 0$, $\mathbb E(X^2) = 1$; $X+Y$ and $X-Y$ are independent; $\phi$ is the characteristic function of $X$ and $Y$ and $ \psi: t \rightarrow \dfrac{\phi(t)}{\phi(-t)}$.
What is, in this case, the series expansion of $\phi$ and $\psi$ at $0$?