On exponentials of exponential generating series

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 $p$.