A new approach to the description of one-parameter groups of formal power series in one indeterminate

Abstract

The aim of the paper is to describe one-parameter groups of formal power series, that is to find a general form of all homomorphisms Θ G : G → Γ, Θ G (t) = ∞ k=1 c k (t)X k , c 1 : G → K \ {0}, c k : G → K for k ≥ 2, from a commutative group (G, +) into the group (Γ, •) of invertible formal power series with coefficients in K ∈ {R, C}. Considering one-parameter groups of formal power series and one-parameter groups of truncated formal power series, we give explicit formulas for the coefficient functions c k with more details in the case where either c 1 = 1 or c 1 takes infinitely many values. Here we give the results much more simply than they were presented in

Families of Commuting Formal Power Series and Formal Functional Equations

“…Iteration groups were studied e.g. in D. N. Lewis [16], S. Sternberg [26], K.-T. Chen [2], St. Scheinberg [25] E. Peschl and L. Reich [18], L. Reich and J. Schwaiger [23] or [24], G. Mehring [17], C. Praagman [19], H. Fripertinger and L. Reich [3,4], and W. Jabłoński [12,13,11,14].…”