Poisson-type processes governed by fractional and higher-order recursive differential equations

We consider some fractional extensions of the recursive differential equation governing the Poisson process, d/dt p(k)(t) = -lambda(p(k)(t) - p(k-1)(t)), k >= 0, t > 0 by introducing fractional time-derivatives of order v, 2v,..., nv. We show that the so-called "Generalized Mittag-Leffler functions" E(alpha,beta)(k)(x), x is an element of R (introduced by Prabhakar [24]) arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the intearrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for t -> infinity. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter v varying in (0, 1]. For integer values of v, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.