With $\varphi \left( p\right) $, $p\geq 0$ the Laplace-Stieltjes transform of some infinitely divisible probability distribution, we consider the solutions to the functional equation \EQN{6}{1}{}{0}{\RD{\CELL{\varphi \left( p\right) =e^{-p\beta }\prod_{i=1}^{m}\varphi ^{\gamma {i}}\left( c{i}p\right) }}{1}{}{}{}}for some $m\geq 1$, $c_{i}>0$, $\gamma _{i}>0$, $i=1,..,m$, $\beta \in \QTR{Bbb}{R}$. We supply its complete solutions in terms of semistable distributions (the ones obtained when $m=1$). We then show how to obtain these solutions as limit laws ($r\uparrow \infty $) of normalized Poisson sums of iid samples when the Poisson intensity $\lambda \left( r\right) $ grows geometrically with $r$.
