Poincar\'e series of modules over compressed Gorenstein local rings

Given positive integers e and s we consider Gorenstein Artinian local rings R of embedding dimension e whose maximal ideal $\mathfrak{m}$ satisfies ${\mathfrak{m}}^s\mathrm{\ne }0={\mathfrak{m}}^{s+1}$. We say that R is a compressed Gorenstein local ring when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If $s\mathrm{\ne }3$, we prove that the Poincare 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 Poincare series exist, due to B{\o}gvad.