Vol. 4, No. 7, 2010

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On exponentials of exponential generating series

Roland Bacher

Vol. 4 (2010), No. 7, 919–942
Abstract

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.

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