Fractional extreme distributions

We consider three classes of linear differential equations on distribution functions, with a fractional order alpha is an element of [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 distribution is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Frechet cases, and with a Le Roy function for the Gumbel case.