  We define and study fractional versions of the well-known Gamma subordinator
$\Gamma :=\{\Gamma (t),$ $t\geq 0\},$ which are obtained by time-changing $%
\Gamma $ by means of an independent stable subordinator or its inverse. Their
densities are proved to satisfy differential equations expressed in terms of
fractional versions of the shift operator (with fractional parameter greater or
less than one, in the two cases). As a consequence, the fractional
generalization of some Gamma subordinated processes (i.e. the Variance Gamma,
the Geometric Stable and the Negative Binomial) are introduced and the
corresponding fractional differential equations are obtained.
