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2010 On exponentials of exponential generating series
Roland Bacher
Algebra Number Theory 4(7): 919-942 (2010). DOI: 10.2140/ant.2010.4.919


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

exp ! : X K [ [ X ] ] 1 + X K [ [ 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.


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Roland Bacher. "On exponentials of exponential generating series." Algebra Number Theory 4 (7) 919 - 942, 2010.


Received: 24 August 2009; Revised: 13 July 2010; Accepted: 17 October 2010; Published: 2010
First available in Project Euclid: 20 December 2017

zbMATH: 1231.11030
MathSciNet: MR2776878
Digital Object Identifier: 10.2140/ant.2010.4.919

Primary: 11B85
Secondary: 11B73 , 11E08 , 11E76 , 22E65

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

Rights: Copyright © 2010 Mathematical Sciences Publishers


Vol.4 • No. 7 • 2010
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