We consider a fractional counting process with jumps of amplitude 1, 2, . . . , k, with k ∈ N, whose probabilities satisfy a suitable system of fractional difference-differential equations. We obtain the moment generating function and the probability law of the resulting process in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When k = 2 we also express the distribution of the first passage time of the fractional counting process in an integral form. Finally, we show that the ratios given by the powers of the fractional Poisson process and of the counting process over their means tend to 1 in probability.

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We investigate the stochastic process defined as the square of the (integrated) symmetric telegraph process. Specifically, we obtain its probability law and a closed form expression of the moment generating function. Some results on the first-passage time through a fixed positive level are also provided. Moreover, we analyze some functionals [Formula: see text] of two independent squared telegraph processes, both in the case [Formula: see text] and [Formula: see text]. Starting from this study, we provide some results on the probability density functions of the two-dimensional radial telegraph process and of the product of two independent symmetric telegraph processes. Some of the expressions obtained are given in terms of new results about derivatives of hypergeometric functions with respect to parameters.

We propose a generalization of the alternating Poisson process from the point of view of fractional calculus. \ud We consider the system of differential equations governing the state probabilities of the alternating Poisson process \ud and replace the ordinary derivative with the fractional derivative (in the Caputo sense). This produces a \ud fractional 2-state point process. We obtain the probability mass function of this process \ud in terms of the (two-parameter) Mittag-Leffler function. \ud Then we show that it can be recovered also by means of renewal theory. \ud We study the limit state probability, and certain proportions involving the fractional moments \ud of the sub-renewal periods of the process. In conclusion, in order to derive new Mittag-Leffler-like \ud distributions related to the considered process, we exploit a transformation acting on \ud pairs of stochastically ordered random variables, which is an extension of the equilibrium operator \ud and deserves interest in the analysis of alternating stochastic processes