Some Compound Fractional Poisson Processes

In this paper, we introduce and study fractional versions of the Bell–Touchard process, Poisson-logarithmic process generalized Pólya–Aeppli process. The state probabilities these compound Poisson processes solve a system differential equations that involves Caputo derivative order 0<β<1. It is shown are limiting cases recently introduced namely, counting We obtain mean, variance, covariance, long-range dependence property, etc., for processes. Further, several equivalent forms one-dimensional distribution