Paradox when expressing an operator in terms of creation/annihilation operators

Question

I'm trying to expand an arbitrary operator using creation/annihilation operators following this post, where $|m\rangle \langle n|$ is expressed as $$ |n\rangle \langle m|~=~\sum_{k\in\mathbb{N}_0} c^{nm}_k (a^{\dagger})^{n+k} a^{m+k}, $$ and $c^{nm}_k$ is solution of $$ \delta_k^0~=~\sum_{r=0}^k c^{nm}_{k-r} \sqrt{ \frac{(n+k)!}{r!} \frac{(m+k)!}{r!} }. $$ But I met a paradox. Let $$ \hat{O}=|m\rangle \langle n|=\sum{C^{mn}_{kl}\hat{a}^{\dagger k}\hat{a}^l}, $$ thus $$ \langle \beta|m\rangle \langle n|\alpha \rangle =\sum{C^{mn}_{kl} \langle \beta| \hat{a}^{\dagger k}\hat{a}^l}|\alpha\rangle $$ where $|\alpha\rangle,|\beta\rangle$ are coherent states. Then $$ \begin{aligned} \mathrm{L}.\mathrm{H}.\mathrm{S}.&=\mathrm{e}^{-\left( |\alpha |^2+|\beta |^2 \right) /2}\frac{\beta ^{*m}\alpha ^n}{\sqrt{m!n!}}\\ =\mathrm{R}.\mathrm{H}.\mathrm{S}.&=\sum{C^{mn}_{kl}\beta ^{*k}\alpha ^l}. \end{aligned} $$ From above equations it can be seen that RHS is an analytic function of $\beta^*$ and $\alpha$ while LHS is not because of the modulus in the exponential. Then we reach the conclusion that $|m\rangle \langle n|$ cannot be expanded by creation/annihilation operators, and the question in the mentioned post still remains: Is it always possible to express an operator in terms of creation/annihilation operators?