A fractional counting process and its connection with the Poisson\n process

We consider a fractional counting process with jumps of amplitude\n$1,2,\\ldots,k$, with $k\\in \\mathbb{N}$, whose probabilities satisfy a suitable\nsystem of fractional difference-differential equations. We obtain the moment\ngenerating function and the probability law of the resulting process in terms\nof generalized Mittag-Leffler functions. We also discuss two equivalent\nrepresentations both in terms of a compound fractional Poisson process and of a\nsubordinator governed by a suitable fractional Cauchy problem. The first\noccurrence time of a jump of fixed amplitude is proved to have the same\ndistribution as the waiting time of the first event of a classical fractional\nPoisson process, this extending a well-known property of the Poisson process.\nWhen $k=2$ we also express the distribution of the first passage time of the\nfractional counting process in an integral form. Finally, we show that the\nratios given by the powers of the fractional Poisson process and of the\ncounting process over their means tend to 1 in probability.\n