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While studying stochastic processes (specifically the paper http://arxiv.org/abs/cond-mat/0412129v1) I have come across a probability distribution that is a generalisation of the standard Poisson distribution. Its mass function is $$P(X=n) = \frac{z^n}{Z} \prod_{i=1}^n w_i^{-1},$$ $$Z = \sum_{n=0}^\infty z^n \prod_{i=1}^n w_i^{-1},$$ where $z$ is the parameter of the distribution and the $w_i$ are a given sequence of positive real numbers.

In the case $w_i=i$, we have $Z = e^z$, and it simplifies to a regular Poisson distribution with parameter $z$.

This seems like quite a natural generalisation of the Poisson distribution but I haven't been able to find any information about (other than in the paper above, which doesn't say much about it). Is this a known distribution and, if so, does it have any nice properties?

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