I have a series that looks like $\sum_{l,m,n}\frac{A^{l}B^{m}C^{n}}{l!m!n!}$ where $A$ is a complex function and $B$ and $C$ are real functions. The summation is finite up to some cutoff $p$. $A$, $B$, and $C$ are small quantities and the series approaches 1 for large $l,m,n$ rather than exploding exponentially. I would like to multiply this summation with its complex conjugate such as \begin{equation} \left[\sum_{l,m,n}\frac{A^{l}B^{m}C^{n}}{l!m!n!}\right]\left[\sum_{l,m,n}\frac{(A^{l})^{*}B^{m}C^{n}}{l!m!n!}\right]. \end{equation} Is there a way to neatly write this summation? Naively, I would like to arrive at something like $\sum_{l,m,n}\left|\frac{A^{l}B^{m}C^{n}}{l!m!n!}\right|^{2}$ but this looks strange after looking at some summation identities.
Edit:
On the other hand, my desired expression does not look too strange for small $A,B,C$ if we consider the triangle identity. Let’s take for instance a single summation over $l$. For simplicity, we start at $l=1$ and end at $p=2$ \begin{equation} \left|\sum_{l}\frac{A^{l}}{l!}\right|^{2}=\left|\frac{A^{1}}{1!}+\frac{A^{2}}{2!}\right|^{2}\equiv\left|f_{1}+f_{2}\right|^{2}\le \left|f_{1}\right|^{2}+\left|f_{2}\right|^{2} +2\left|f_{1}\right|\left|f_{2}\right| \end{equation} where from the previous descriptions $f_{1}\gt f_{2}$ in general. It follows that for large $p$ \begin{equation} \left|f_{1}+f_{2}+\ldots+f_{p}\right|^{2}\le \left|f_{1}\right|^{2}+\left|f_{2}\right|^{2}+\ldots+\left|f_{p}\right|^{2} +p\left|f_{1}\right|\left|f_{2}\right|\ldots\left|f_{p}\right| \end{equation} the last term can be ignored for small $A$. Therefore, \begin{equation} \left|\sum_{l}^{p}f_{l}\right|^{2}\approx \sum_{l}^{p} \left|f_{l}\right|^{2} \end{equation} for large $p$ and small $A$.
