\(q\)-analogues of the hyper-Poisson distribution (Q1347975)

This article investigates two \(q\)-analogues of the hyper-Poisson distribution, motivated by the application of the `displaced Poisson' procedure to Heine and Euler analogues of the Poisson distribution. Several properties of the two considered hyper-Poisson distributions, the `displaced-Heine' and the `displaced-Euler' one, are considered: limits to a Bernoulli distribution, recurrence relations for their probabilities, expression of the distribution modes. There are described the models involving mixtures of other \(q\)-series distributions and group-size models, the latter ones being interpreted as birth-death models and as random walk models.