  In this paper, we introduce and study fractional versions of three compound
Poisson processes, namely, the Bell-Touchard process, the Poisson-logarithmic
process and the generalized P\'olya-Aeppli process. It is shown that these
processes are limiting cases of a recently introduced process by Di Crescenzo
et al. (2016), namely, the generalized counting process. We obtain the mean,
variance, covariance, long-range dependence property etc. for these processes.
Also, we obtain several equivalent forms of the one-dimensional distribution of
fractional versions of these processes.
