#### 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

${exp}_{!}:X\mathbb{K}\left[\left[X\right]\right]\to 1+X\mathbb{K}\left[\left[X\right]\right]$

for the associated Lie group with multiplication given by the shuffle product is well-defined over an arbitrary field $\mathbb{K}$ 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 $X\mathbb{K}\left[\left[X\right]\right]$ and the subgroup of rational, respectively algebraic series in the group $1+X\mathbb{K}\left[\left[X\right]\right]$ endowed with the shuffle product, if the field $\mathbb{K}$ is a subfield of the algebraically closed field ${\overline{\mathbb{F}}}_{p}$ 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