Foundations of Computational Mathematics

, Volume 9, Issue 3, pp 295–316 | Cite as

A Magnus- and Fer-Type Formula in Dendriform Algebras

  • Kurusch Ebrahimi-FardEmail author
  • Dominique Manchon
Open Access


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.


Linear differential equation Linear integral equation Magnus expansion Fer expansion Dendriform algebra Pre-Lie algebra Rota–Baxter algebra Binary rooted trees 

Mathematics Subject Classification (2000)

16W25 17A30 17D25 37C10 05C05 81T15 


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Copyright information

© SFoCM 2008

Authors and Affiliations

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

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