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Recently the so-called Prabhakar generalization of the fractional Poissoncounting process attracted much interest for his flexibility to adapt realworld situations. In this renewal process the waiting times between events areIID continuous random variables. In the present paper we analyze discrete-timecounterparts: Renewal processes with integer IID interarrival times whichconverge in well-scaled continuous-time limits to the Prabhakar-generalizedfractional Poisson process. These processes exhibit non-Markovian features andlong-time memory effects. We recover for special choices of parameters thediscrete-time versions of classical cases, such as the fractional Bernoulliprocess and the standard Bernoulli process as discrete-time approximations ofthe fractional Poisson and the standard Poisson process, respectively. Wederive difference equations of generalized fractional type that govern thesediscrete time-processes where in well-scaled continuous-time limits knownevolution equations of generalized fractional Prabhakar type are recovered. Wealso 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 versionof Prabhakar renewal process. We derive the generalized fractionaldiscrete-time Kolmogorov-Feller difference equations governing the resultingstochastic motion. Prabhakar-discrete-time processes open a promising fieldcapturing several aspects in the dynamics of complex systems.

On Discrete Time Prabhakar-Generalized Fractional Poisson Processes and Related Stochastic Dynamics