Lagrange inversion and combinatorial species with uncountable color palette

TL;DR

This paper extends the Lagrange-Good inversion formula to multivariate functionals with uncountably many variables, exploring its relation to tree-based inversion methods and clarifying the underlying cancellations and connections in various applications.

Contribution

It introduces a multivariate Lagrange-Good formula for uncountably many variables and analyzes its relationship with tree-based inversion formulas, highlighting new theoretical insights.

Findings

Established a multivariate Lagrange-Good formula for uncountable variables

Clarified the cancellations between different inversion formulas

Connected the approach to various application contexts

Abstract

We prove a multivariate Lagrange-Good formula for functionals of uncountably many variables and investigate its relation with inversion formulas using trees. We clarify the cancellations that take place between the two aforementioned formulas and draw connections with similar approaches in a range of applications.