What is a good way of summarizing when "generating function" approach is useful to an audience of practitioners?

I'm giving a talk on training neural networks (see [Velikanov, Kuznedelev, and Yarotsky - A view of mini-batch SGD via generating functions: conditions of convergence, phase transitions, benefit from negative momenta](https://arxiv.org/abs/2206.11124)), and want to motivate the idea of generating functions to audience not exposed to this concept.

One example I was considering was this:
consider following matrix sequences

\begin{gather*}
A^0,A^1,A^2,\dotsc\\
B^0,B^1,B^2\dotsc.\\
\end{gather*}

We can consider a new sequence $(A+B)^n$
$$(A+B)^0,(A+B)^1,(A+B)^2,\dotsc.$$

If the original sequences had generating functions $G_A$, $G_B$, the combined sequence's generating function is:

$$G_{A+B}=(I-x G_A B)^{-1} G_A.$$

In this case, the "generating function" approach replaces matrix powering with inversion, so it helps when inversion is easy (i.e., $A+B$ is diagonal + rank 1), and doesn't help if matrix powering is easy.