Vol. 4, No. 7, 2010

Download this article
Download this article For screen
For printing
Recent Issues

Volume 12, 1 issue

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors' Addresses
Editors' Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
On exponentials of exponential generating series

Roland Bacher

Vol. 4 (2010), No. 7, 919–942

After identification of the algebra of exponential generating series with the shuffle algebra of ordinary formal power series, the exponential map

exp! : XK[[X]]1 + XK[[X]]

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 XK[[X]] and the subgroup of rational, respectively algebraic series in the group 1 + XK[[X]] endowed with the shuffle product, if the field K is a subfield of the algebraically closed field F¯p of characteristic p.

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
Received: 24 August 2009
Revised: 13 July 2010
Accepted: 17 October 2010
Published: 29 January 2011
Roland Bacher
Université Grenoble I
Institut Fourier
100, rue des Maths
Boîte Postale 74
38402 St. Martin d’Hères