, Volume 9, Issue 3, pp 295-316,
Open Access This content is freely available online to anyone, anywhere at any time.

A Magnus- and Fer-Type Formula in Dendriform Algebras

Abstract

We provide a refined approach to the classical Magnus (Commun. Pure Appl. Math. 7:649–673, [1954]) and Fer expansion (Bull. Classe Sci. Acad. R. Belg. 44:818–829, [1958]), 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+λ a X and Y=1−λ Y a in A[[λ]] 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.

Communicated by Arieh Iserles.