Combinatorial generating functions encode enumerative information about a class of discrete objects. The functional equations they satisfy can provide key insight on the structure of the objects. This talk examines connections between combinatorial structure and the differential transcendence of related series.

In particular, I will focus on recent work with Lucia Di Vizio and Gwladys Fernandes characterizing series solutions of order 1 iterative solutions. A common dichotomy arises: solutions are either differentially transcendental or algebraic. The proof exploits differential Galois theory. I will then mention an application done with Yakob Kahane, determining the nature of Green’s functions of a well-known class of fractal graphs.

Given the structure that is slowly being revealed, this combinatorial framework may be an interesting candidate for a more model theoretic treatment.