For about two centuries, the Poisson process with the memoryless property of the corresponding exponential distribution has been used as the simplest and one of the most important stochastic models, while there are many processes with long memory in the world around us. Therefore, it is useful to generalize the Poisson process to include these processes. In this generalization, a parameter $alin (0, 1]$ is added as the fractional power of the process. In this paper, the transition from the ordinary Poisson process to its fractional generalization, namely the fractional Poisson process ($f$), is investigated and the relationship of $f$ with $alpha$-stable distributions is proven by solving an integral equation. Finally, the characteristics of these estimators are tested using simulation data.