## On exponentials of exponential generating series.(English)Zbl 1231.11030

The shuffle product of two formal power series $$\sum_{n=0}^{\infty}{\alpha_nX^n}$$ and $$\sum_{n=0}^{\infty}{\beta_n X^n}$$ with coefficients in a field $$\mathbb K$$ is defined by $\sum_{n=0}^{\infty}{\gamma_nX^n}:=\sum_{n=0}^{\infty}{\alpha_nX^n} \operatorname{\text Ш} \sum_{n=0}^{\infty}{\beta_nX^n}$ where $\gamma_n=\sum_{n=0}^{\infty}{\binom{n}{k}\alpha_k \beta_{n-k}}.$ A result which goes back to Hurwitz shows that the additive group $$(X\mathbb K [[X]],+)$$ and the shuffle group $$(1+X\mathbb K [[X]], \operatorname{\text Ш} )$$ are isomorphe. The map which gives this isomorphism is the exponential map defined as follows. If $$A=\sum_{n=1}^{\infty}{\alpha_n X^n}$$ and $$B=\sum_{n=1}^{\infty}{\beta_n X^n}$$, then $\exp_{!}(A):=1+B$ when $\exp(\sum_{n=1}^{\infty}{\frac{\alpha_n}{n!} X^n})=1+\sum_{n=1}^{\infty}{\frac{\beta_n}{n!} X^n}.$
In this article, the author proves that the map $$\exp_{!}$$ also induces a group isomorphism between the subgroup of rational (respectively algebraic) series of $$(X\mathbb K [[X]], +)$$ and the subgroup of rational (respectively algebraic) series of $$(1+X\mathbb K [[X]], \operatorname{\text Ш} )$$, if $$\mathbb K$$ is a subfield of the algebraically closed field $$\overline{\mathbb F}_p$$ of positive characteristic $$p$$. The author also shows that this result is not true if the field is of characteristic zero.

### MSC:

 11B85 Automata sequences 11B73 Bell and Stirling numbers 11E08 Quadratic forms over local rings and fields 11E76 Forms of degree higher than two 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
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