A Magnus- and Fer-Type Formula in Dendriform Algebras
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- 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, ) and Fer expansion (Bull. Classe Sci. Acad. R. Belg. 44:818–829, ), 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.