# Families of commuting formal power series and formal functional equations

### Abstract

In this paper we describe families of commuting invertible formal power series in one indeterminate over ℂ, using the method of formal functional equations. We give a characterization of such families where the set of multipliers (first coefficients) σ of its members F(x) = σx+... is infinite, in particular of such families which are maximal with respect to inclusion, so called families of *type* I. The description of these families is based on Aczél–Jabotinsky differential equations, iteration groups, and on some results on normal forms of invertible series with respect to conjugation.

### Keywords

commuting formal power series; maximal families of commuting formal power series; maximal abelian subgroups in $\Gamma$; formal functional equations; formal partial differential equations; Aczél–Jabotinsky equation; Briot–Bouquet equation; formal iteration groups of type I; ring of formal power series over $\mathbb{C}$

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*Annales Mathematicae Silesianae*,

*35*(1), 55-76. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/13474

Institute of Mathematics and Scientific Computing, University of Graz, Austria Austria

https://orcid.org/0000-0001-7449-8532

Institute of Mathematics and Scientific Computing, University of Graz, Austria Austria

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