Fractional discrete processes: compound and mixed Poisson representations

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We consider two fractional versions of a family of nonnegative integer valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Polya-Aeppli, the Poisson Inverse Gaussian and the Negative Binomial. We also define and study some more general fractional versions with two fractional parameters.