Explore a comprehensive lecture on functional relations between generating series of cumulants and moments in higher order free probability. Delve into the formulation and proof overview of these formulas, examining the major steps and techniques employed. Gain insights into recent developments and the universal role of these functional relations in matrix models and topological recursion theory. Learn from Sergey Shadrin's expertise, drawing upon his collaborative research papers with renowned mathematicians in the field. Understand the connections between non-commutative geometry and topological recursion as presented in this workshop talk from the Erwin Schrödinger International Institute for Mathematics and Physics.