  Recently the so-called Prabhakar generalization of the fractional Poisson
counting process attracted much interest for his flexibility to adapt real
world situations. In this renewal process the waiting times between events are
IID continuous random variables. In the present paper we analyze discrete-time
counterparts: Renewal processes with integer IID interarrival times which
converge in well-scaled continuous-time limits to the Prabhakar-generalized
fractional Poisson process. These processes exhibit non-Markovian features and
long-time memory effects. We recover for special choices of parameters the
discrete-time versions of classical cases, such as the fractional Bernoulli
process and the standard Bernoulli process as discrete-time approximations of
the fractional Poisson and the standard Poisson process, respectively. We
derive difference equations of generalized fractional type that govern these
discrete time-processes where in well-scaled continuous-time limits known
evolution equations of generalized fractional Prabhakar type are recovered. We
also develop in Montroll-Weiss fashion the `Prabhakar Discrete-time random walk
(DTRW)' as a random walk on a graph time-changed with a discrete-time version
of Prabhakar renewal process. We derive the generalized fractional
discrete-time Kolmogorov-Feller difference equations governing the resulting
stochastic motion. Prabhakar-discrete-time processes open a promising field
capturing several aspects in the dynamics of complex systems.
