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