Recursive characterization of a large family of discrete probability distributions showing extra-Poisson variation

This article analyses the broad family of discrete probability distributions generated by relating Prob (y) to Prob (y?1), …, Prob (y?n), for some n≥1, through a recursive equation. This family contains the binomial, negative binomial and Poisson distributions as well as the Katz family of distributions. In addition, the suggested family contains some convolutions of Poisson distributions and other generalized distributions, which provide models for Poisson overdispersion or underdispersion.