A Magnus- and Fer-type formula in dendriform algebras

TL;DR

This paper introduces a new approach to Magnus and Fer expansions using dendriform and pre-Lie algebra structures, providing recursive formulas and infinite product solutions with several applications.

Contribution

It presents a refined, algebraic framework for classical expansions, connecting dendriform and pre-Lie algebras to solve specific equations in a novel way.

Findings

Recursive formulas for logarithms of solutions in dendriform algebras.

Infinite product expansions of solutions as exponentials.

Applications demonstrating the utility of the new formulas.

Abstract

We provide a refined approach to the classical Magnus and Fer expansion, unveiling a new structure by using the language of dendriform and pre-Lie algebras. The recursive formula for the logarithm of the solutions of the equations X=1+ta<X and Y=1-tY> a in A[[t]] is provided, where (A,<,>) is a dendriform algebra. Then, we present the solutions to these equations as an infinite product expansion of exponentials. Both formulae involve the pre-Lie product naturally associated with the dendriform structure. Several applications are presented.