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The complete set of solutions can be written under of the form $$\psi_n(\phi)=A_ne^{in\phi}+B_ne^{-in\phi},\quad n\in\mathbb{N}$$ or equivalently under the form $$\psi_n'(\phi)=C_ne^{in\phi},\quad n\in\mathbb{Z}$$ Note that $\psi_n\ne\psi_n'$ but, in both cases, the general solution is $$\psi(\phi)=\sum_{n\in\mathbb{Z}}C_ne^{in\phi}$$ with $C_n=A_n$ if $n>0$, $C_n=B_{-n}$ if $n<0$, and $C_0=A_0+B_0$.