One-parameter Groups of Formal Power Series of One Indeterminate

Abstract

The aim of the paper is to give a survey of results and open problems concerning one-parameter groups of formal power series in one indeterminate. A one-parameter group of formal power series is a homomorphism $$G \ni t\mapsto \sum\limits_{k=1}^{\infty }{c}_{ k}(t){X}^{k} \in {\Gamma }^{\infty }$$ of some group G into the group Γ ∞ of invertible formal power series. To describe arbitrary one-parameter groups, several tools from different branches of mathematics are used (differential equations, functional equations, abstract algebra). Unfortunately, the problem of to give description of one-parameter group of formal power series for arbitrary abelian group G seems to be still open. Independent of these problems, in ring of formal power series, homomorphisms of some groups into differential groups L s 1 and L ∞ 1 have been examined. Tools used there are rather simple and did not exceed simple methods of substitution and changing variables adopted form the theory of functional equations. Consequently, only some homomorphisms into groups L s 1 for s≤5 are known and a partial result on homomorphisms into L s 1 with arbitrary s is proved. It appears that all the results on one-parameter groups can be transferred onto differential groups L s 1 and L ∞ 1. This can be done using algebraic isomorphisms between Γ ∞ and L ∞ 1 as well as between a group of invertible truncated formal power series and L s 1.KeywordsOne-parameter groupTranslation equationFormal power series