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), 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.
