On exponentials of exponential generating series

for the associated Lie group with multiplication given by the shuffle product is well-defined over an arbitrary field $K$ by a result going back to Hurwitz. The main result of this paper states that ${exp}_{!}$ and its reciprocal map ${log}_{!}$ induce a group isomorphism between the subgroup of rational, respectively algebraic series of the additive group $X K [[X]]$ and the subgroup of rational, respectively algebraic series in the group $1 + X K [[X]]$ endowed with the shuffle product, if the field $K$ is a subfield of the algebraically closed field ${\bar{F}}_{p}$ of characteristic $p$ .

Keywords

Bell numbers, exponential function, shuffle product, formal power series, divided powers, rational series, algebraic series, homogeneous form, automaton sequence

Mathematical Subject Classification 2000

Primary: 11B85

Secondary: 11B73, 11E08, 11E76, 22E65

Milestones

Received: 24 August 2009

Revised: 13 July 2010

Accepted: 17 October 2010

Published: 29 January 2011

Authors

Roland Bacher
Université Grenoble I CNRS UMR 5582 Institut Fourier 100, rue des Maths Boîte Postale 74 38402 St. Martin d’Hères France

Vol. 4, No. 7, 2010

Roland Bacher

Abstract

Keywords

Mathematical Subject Classification 2000

Milestones

Authors