  We consider three classes of linear differential equations on distribution
functions, with a fractional order $\alpha\in [0,1].$ The integer case $\alpha
=1$ corresponds to the three classical extreme families. In general, we show
that there is a unique distribution function solving these equations, whose
underlying random variable is expressed in terms of an exponential random
variable and an integral transform of an independent $\alpha-$stable
subordinator. From the analytical viewpoint, this law is in one-to-one
correspondence with a Kilbas-Saigo function for the Weibull and Fr\'echet
cases, and with a Le Roy function for the Gumbel case. By the stochastic
representation, we can derive several analytical properties for the latter
special functions, extending known features of the classical Mittag-Leffler
function, and dealing with monotonicity, complete monotonicity, infinite
divisibility, asymptotic behaviour at infinity, uniform hyperbolic bounds.
