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. 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?