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 $C_0=A_0+B_0$.

The complete set of solutions of the azimutal equation can be written under of the form $$\psi_m(\phi)=A_me^{im\phi}+B_me^{-im\phi},\quad m\in\mathbb{N}$$ or equivalently under the form $$\psi_m'(\phi)=C_ne^{im\phi},\quad m\in\mathbb{Z}$$ Note that $\psi_m\ne\psi_m'$ but, in both cases, the general solution is $$\psi(\phi)=\sum_{m\in\mathbb{Z}}C_me^{im\phi}$$ with $C_m=A_m$ if $m>0$, $C_m=B_{-m}$ if $m<0$, and $C_0=A_0+B_0$.

Similarly, the solutions of the full Schrödinger equation are of the form $$\psi_{n,l,m}(r,\theta,\phi)=R_{nl}(r)P_{lm}(\cos\theta)e^{im\phi}$$ so the general solution is $$\psi(r,\theta,\phi)=\sum_{n,l,m} D_{n,l,m}R_{nl}(r)P_{lm}(\cos\theta)e^{im\phi}$$ where the sum over $m$ extends over $\mathbb{Z}$. Since $P_{lm}(\cos\theta)$ is proportionnal to $P_{l,-m}(\cos\theta)$ (https://en.wikipedia.org/wiki/Associated_Legendre_polynomials), you can also find some coefficients $F_{n,l,m}$ and $G_{n,l,m}$ such that $$\psi(r,\theta,\phi)=\sum_{n,l,m} R_{nl}(r)P_{lm}(\cos\theta)\big[F_{n,l,m}e^{im\phi}+G_{n,l,m}e^{-im\phi}\big]$$ where the sum over $m$ extends now only over $\mathbb{N}$.