On exponentials of exponential generating series
Roland Bacher · Algebra & Number Theory · 2010
After identification of the algebra of exponential generating series with the shuffle algebra of ordinary formal power series, the exponential mapfor the associated Lie group with multiplication given by the shuffle product is well-defined over an arbitrary field ދ 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 [[ދX ]] and the subgroup of rational, respectively algebraic series in the group 1 + X [[ދX ]] endowed with the shuffle product, if the field ދ is a subfield of the algebraically closed field ކ p of characteristic p.