  We consider a fractional counting process with jumps of amplitude
$1,2,\ldots,k$, with $k\in \mathbb{N}$, 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.
