Rooted trees and an exponential-like series

TL;DR

This paper explores generalized power series linked to augmented operads, particularly the PreLie operad, and introduces the PreLie exponential derived from flow equations on vector fields.

Contribution

It introduces the PreLie exponential as a novel element in the group of generalized power series associated with the PreLie operad.

Findings

Definition of the PreLie exponential within the operad framework

Connection between flow equations and exponential elements

Extension of operad-based series to the PreLie context

Abstract

This paper deals with a group of generalized power series associated to any augmented operad, focusing on the case of the PreLie operad. The solution of flow equations using the pre-Lie structure on vector fields on an affine space gives rise to an interesting element of this group, which deserves to be called the PreLie exponential.