Displaced Poisson distribution

In statistics, the displaced Poisson, also known as the hyper-Poisson distribution, is a generalization of the Poisson distribution. The probability mass function is

where and r is a new parameter; the Poisson distribution is recovered at r = 0. Here is the incomplete gamma function and s is the integral part of r. The motivation given by Staff is that the ratio of successive probabilities in the Poisson distribution (that is ) is given by for and the displaced Poisson generalizes this ratio to .