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 $\bar{\mathbb F}_p$ of characteristic $p$.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2008
- DOI:
- 10.48550/arXiv.0807.0540
- arXiv:
- arXiv:0807.0540
- Bibcode:
- 2008arXiv0807.0540B
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Combinatorics
- E-Print:
- 25 pages