# On exponentials of exponential generating series

Abstract : Identifying the algebra of exponential generating series with the shuffle algebra of formal power series, one can define an exponential map ${\mathop{exp}}_!:X\mathbb K[[X]]\longrightarrow 1+X\mathbb K[[X]]$ for the associated Lie group formed by exponential generating series with constant coefficient $1$ over an arbitrary field $\mathbb K$. The main result of this paper states that the map ${\mathop{exp}}_!$ (and its inverse map ${\mathop{log}}_!$) induces a bijection between rational, respectively algebraic, series in $X\mathbb K [[X]]$ and $1+X\mathbb K[[X]]$ if the field $\mathbb K$ is a subfield of the algebraically closed field $\overline{\mathbb F}_p$ of characteristic $p$.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-00292997
Contributor : Roland Bacher <>
Submitted on : Monday, October 18, 2010 - 4:08:46 PM
Last modification on : Thursday, January 11, 2018 - 6:12:13 AM
Document(s) archivé(s) le : Wednesday, January 19, 2011 - 2:50:49 AM

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expexp1.pdf
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### Identifiers

• HAL Id : hal-00292997, version 4
• ARXIV : 0807.0540

### Citation

Roland Bacher. On exponentials of exponential generating series. Algebra & Number Theory, 2010, 4 (7), pp.919-942. ⟨hal-00292997v4⟩

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