  We are interested in the differential equations satisfied by the density of
the Geometric Stable processes
$\mathcal{G}_{\alpha}^{\beta}=\left\{\mathcal{G}_{\alpha}^{\beta}(t);t\geq
0\right\} $, with stability \ index $% \alpha \in (0,2]$ and asymmetry
parameter $\beta \in \lbrack -1,1]$, both in the univariate and in the
multivariate cases. We resort to their representation as compositions of stable
processes with an independent Gamma subordinator. As a preliminary result, we
prove that the latter is governed by a differential equation expressed by means
of the shift operator. As a consequence, we obtain the space-fractional
equation satisfied by the density of $\mathcal{G}_{\alpha}^{\beta}.$ For some
particular values of $% \alpha $ and $\beta ,$ we get some interesting results
linked to well-known processes, such as the Variance Gamma process and the
first passage time of the Brownian motion.
