Consider a discrete random variable $N\in\mathbb N$ with 

- $\mathbb P(N=0) = p$,
- $\mathbb P(N=n) = (1-p)(1-q)q^n$ for $n\neq 0$.


Then the probability generating function of $N$ 
$$\mathbb E(z^N) = \frac{p + (1-p-q)z}{1-qz}$$  is a Mobius transform.


It's pretty easy to show that these are all the distributions with Mobius PGF's
They come up in birth-death processes see for example [Kendal 1958][1].
They have a few nice properties related to the fact that the Mobius transformations form a group under composition.


I'm using them to simplify a few calculations and I was wondering if they had a name.
Has anyone come across a reference where these things are treated explicitly?


[1]: https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-19/issue-1/On-the-Generalized-Birth-and-Death-Process/10.1214/aoms/1177730285.full