Foundations of Computational Mathematics

, Volume 9, Issue 3, pp 295–316

A Magnus- and Fer-Type Formula in Dendriform Algebras

Open AccessArticle

DOI: 10.1007/s10208-008-9023-3

Cite this article as:
Ebrahimi-Fard, K. & Manchon, D. Found Comput Math (2009) 9: 295. doi:10.1007/s10208-008-9023-3


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+λaX and Y=1−λYa 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.


Linear differential equationLinear integral equationMagnus expansionFer expansionDendriform algebraPre-Lie algebraRota–Baxter algebraBinary rooted trees

Mathematics Subject Classification (2000)

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© SFoCM 2008

Authors and Affiliations

  1. 1.Max-Planck-Institut für MathematikBonnGermany
  2. 2.Université Blaise PascalAubièreFrance