Sur l'Inversion et l'Iteration Continue des Séries Formelles

This paper deals with the composition of normalised formal power series, in one variable, over an arbitrary field K of characteristic zero. A suitable group structure B⊙ on the set B of polynomial sequences of binomial type is introduced. This group is used first to obtain many formal variants of the classical Lagrange inversion formula (without using any complex integration). Secondly, via one-parameter subgroups of B⊙, iteration (i.e., successive composition) of normalised formal power series is studied in detail for arbitrary orders s∈K (“continuous” iteration). The case s = −1 coincides with power series inversion. Many new formulas are derived in the course of the text. The end of the work contains suggestions for future research