  This paper defines a new class of fractional differential operators alongside
a family of random variables whose density functions solve fractional
differential equations equipped with these operators. These equations can be
further used to construct fractional integro-differential equations for the
ruin probabilities in collective renewal risk models, with inter-arrival time
distributions from the aforementioned family. Gamma-time risk models and
fractional Poisson risk models are two specific cases among them, whose ruin
probabilities have explicit solutions, when claim sizes distributions exhibit
rational Laplace transforms.
