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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. Theirdensities are proved to satisfy differential equations expressed in terms offractional versions of the shift operator (with fractional parameter greater orless than one, in the two cases). As a consequence, the fractionalgeneralization of some Gamma subordinated processes (i.e. the Variance Gamma,the Geometric Stable and the Negative Binomial) are introduced and thecorresponding fractional differential equations are obtained.

Fractional Gamma process and fractional Gamma-subordinated processes