  In this paper, we obtain additional results for a fractional counting process
introduced and studied by Di Crescenzo et al. (2016). For convenience, we call
it the generalized fractional counting process (GFCP). It is shown that the
one-dimensional distributions of the GFCP are not infinitely divisible. Its
covariance structure is studied using which its long-range dependence property
is established. It is shown that the increments of GFCP exhibits the
short-range dependence property. Also, we prove that the GFCP is a scaling
limit of some continuous time random walk. A particular case of the GFCP,
namely, the generalized counting process (GCP) is discussed for which we obtain
a limiting result, a martingale result and establish a recurrence relation for
its probability mass function. We have shown that many known counting processes
such as the Poisson process of order $k$, the P\'olya-Aeppli process of order
$k$, the negative binomial process and their fractional versions etc. are other
special cases of the GFCP. An application of the GCP to risk theory is
discussed.
