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 $p$.