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 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 Jabłoński and Reich (Abh. Math. Sem. Univ. Hamburg 75:179–201, 2005; Result Math 47:61–68, 2005; Publ Math Debrecen 73(1–2):25–47, 2008). Also the case im c 1 = E m (here E m 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, 2005; Result Math 47:61–68, 2005; Publ Math Debrecen 73(1–2):25–47, 2008), will be discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Fejdasz B., Wilczyński Z.: On some s-1—parameter subsemigroups of the group \({L_s^1}\) . Zeszyty Naukowe WSP w Rzeszowie 1(2), 45–50 (1990)
Henrici, P.: Applied and computational complex analysis, vol. I. Power series—integration—conformal mapping—location of zeros. Wiley, New York (1974)
Hille, E.: Ordinary differential equations in the complex domain. In: Pure and Aapplied Mathematics. Wiley, New York (1976)
Jabłoński W.: On some subsemigroups of the group \({L_s^1}\) . Rocznik Nauk.–Dydak. WSP w Krakowie 14(189), 101–119 (1997)
Jabłoński W.: On extensibility of some homomorphisms. Rocznik Nauk.-Dydak. WSP w Krakowie 16(207), 35–43 (1999)
Jabłoński W., Reich L.: On the solutions of the translation equation in rings of formal power series. Abh. Math. Sem. Univ. Hamburg 75, 179–201 (2005)
Jabłoński W., Reich L.: On the form of homomorphisms into the differential group \({L_s^1}\) and their extensibility. Result. Math. 47, 61–68 (2005)
Jabłoński W., Reich L.: On the standard form of the solution of the translation equation in rings of formal power series. Math. Panon. 18(2), 169–187 (2007)
Jabłoński W., Reich L.: On homomorphisms of an abelian group into the group of invertible formal power series. Publ. Math. Debrecen 73(1-2), 25–47 (2008)
Lang, S.: Algebra. Addison-Wesley, Reading (1965)
Reich, L.: On the local distribution of iterable power series transformations in one indeterminate. In: Functional analysis, III (Dubrovnik, 1989), pp. 307–323. Various Publ. Ser. (Aarhus), vol. 40. Aarhus Univ., Aarhus (1992)
Reich L., Schwaiger J.: Über analytische Iterierbatkeit formaler Potenzreihenvektoren. Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II 184(8-10), 599–617 (1975)
Reich L., Schwaiger J.: Über einen Satz von Shl. Sternberg in der Theorie der analytischen Iterationen. Monatshefte fÜr Mathematik 83, 207–221 (1977)
Scheinberg S.: Power series in one variable. J. Math. Anal. Appl. 31, 321–333 (1970)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Jabłoński, W., Reich, L. A new approach to the description of one-parameter groups of formal power series in one indeterminate. Aequat. Math. 87, 247–284 (2014). https://doi.org/10.1007/s00010-013-0232-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-013-0232-8