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.


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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