The Poincare' series of modules over compressed Gorenstein local rings

Given two positive integers $e$ and $s$ we consider Gorenstein Artinian local rings $R$ whose maximal ideal $\fm$ satisfies $\fm^s\ne 0=\fm^{s+1}$ and $\rank_{R/\fm}(\fm/\fm^2)=e$. We say that $R$ is a {\it compressed Gorenstein local ring} when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If $s\ne 3$, we prove that the Poincar\'e series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When $s$ is even this formula depends only on the integers $e$ and $s$. Note that for $s=3$ examples of compressed Gorenstein local rings with transcendental Poincar\'e series exist, due to B\o gvad