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