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- Title
A new approach to the description of one-parameter groups of formal power series in one indeterminate.
- Authors
Jabłoński, Wojciech; Reich, Ludwig
- 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 $${\Theta_G : G \to \Gamma}$$ , $${\Theta_G(t) = \sum_{k=1}^{\infty} c_k(t)X^k}$$ , $${c_1 : G \to \mathbb{K} \setminus\{0\}}$$ , $${c_k : G \to \mathbb{K}}$$ for k ≥ 2, from a commutative group ( G, + ) into the group $${(\Gamma, \circ)}$$ of invertible formal power series with coefficients in $${\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}}$$ . 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 with more details in the case where either c = 1 or c takes infinitely many values. Here we give the results much more simply than they were presented in Jabłoński and Reich (Abh. Math. Sem. Univ. Hamburg 75:179-201, ; Result Math 47:61-68, ; Publ Math Debrecen 73(1-2):25-47, ). Also the case im c = E (here E stands for the group of all complex roots of order m of 1), not considered in Jabłoński and Reich (Abh. Math. Sem. Univ. Hamburg 75:179-201, ; Result Math 47:61-68, ; Publ Math Debrecen 73(1-2):25-47, ), will be discussed.
- Subjects
POWER series; HOMOMORPHISMS; ABELIAN groups; P-adic logarithms; DIFFERENTIAL equations; FUNCTIONAL equations; ITERATIVE methods (Mathematics); DIOPHANTINE equations
- Publication
Aequationes Mathematicae, 2014, Vol 87, Issue 3, p247
- ISSN
0001-9054
- Publication type
Article
- DOI
10.1007/s00010-013-0232-8