99 Laplace transforms which are strictly proper rationals

Let $y(t)$ be a function of the form

$$y(t)=\sum _{j=1}^n{a}_{j,n_j}{t}^{n_j}{e}^{{\lambda }_{j}t}.$$

Then the Laplace transform $Y$ of $y$ is a strictly proper rational function with partial fraction decomposition

$$Y(s)=\sum _{j=1}^n\frac{n_j!a_{j,n_j}}{{(s-{\lambda }_j)}^{n_j+1}}.$$

Conversely, all strictly proper rational functions arise thus.