As far as I know, it seems that a general Laplace transform of the product of two functions, like $\mathscr{L}\left\{f(t) g(t) \right\}$ is not possible. Then, if $f(t)$ is a known function with the following expression \begin{equation} f(t) = \frac{\sum\limits_{n=1}^\infty A_n \text{e}^{k_n t}}{\sum\limits_{n=1}^\infty B_n \text{e}^{k_n t}}, \end{equation} where $A_n$, $B_n$ and $k_n$ are known coefficients, and not functions of $t$.
In this case, is it possible to work out the explicit Laplace transform of $f(t) g(t)$ in terms of $\hat{g}(s)$ and some other functions of $s$?