The goal of my talk is to give a formulation and, mostly important, an overview of the proof (I plan to explain the big steps and techniques used there) of the formulas that give functional relations between generating series of cumulants and moments in higher order free probability. If time permits, I'll explain some recent further developments and a universal role of these functional relations in the theory of matrix models and topological recursion.
Along the lines of my papers
2112.12184 (with Gaëtan Borot, Séverin Charbonnier, Elba Garcia-Failde, Felix Leid),
2106.08368 (with Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian),
and 2212.00320 (with Alexander Alexandrov, Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian).