An alternative formulation of the discrete-time fractional Poisson process

This paper introduces a discrete-time fractional Poisson process defined as a renewal process, where the waiting times follow a discrete Mittag-Leffler distribution. We investigate its fundamental properties by explicitly deriving the probability generating function of the waiting times and the exact probability distribution of the event counts. Through this analysis, we reveal that, unlike its continuous-time counterpart, our renewal-based model is not mathematically equivalent to the process constructed via subordination using the Sibuya distribution.