The tridendriform structure of a Magnus expansion

TL;DR

This paper explores the algebraic and combinatorial structures underlying the Magnus expansion, introducing a tridendriform algebra framework to derive new formulas for solutions of linear difference equations.

Contribution

It develops a novel tridendriform algebra approach to the Magnus expansion, providing explicit formulas using planar reduced trees and word quasi-symmetric functions.

Findings

Derived a closed formula for the Magnus expansion in free tridendriform algebra

Established a discrete analogue of the Mielnik-Plebanski-Strichartz formula

Connected algebraic structures with solutions of linear difference equations

Abstract

The notion of trees plays an important role in Butcher's B-series. More recently, a refined understanding of algebraic and combinatorial structures underlying the Magnus expansion has emerged thanks to the use of rooted trees. We follow these ideas by further developing the observation that the logarithm of the solution of a lihear first-order finite-difference equation can be written in terms of the Magnus expansion taking place in a pre-Lie algebra. By using basic combinatorics on planar reduced trees we derive a closed formula for the Magnus expansion in the context of free tridendriform algebra. The tridendriform algebra structure on word quasi-symmetric functions permits us to derive a discrete analogue of the Mielnik-Plebanski-Strichartz formula for this logarithm.