Exponential functions in prime characteristic

Summary.

In this note we determine all power series

$$ F(X) \in 1 + X{{\mathbb{F}}}_{p} {[}{[}X{]}{]} $$

such that (F(X + Y))⁻¹F(X)F(Y) has only terms of total degree a multiple of p. Up to a scalar factor, they are all the series of the form F(X)  =  E _p (cX)· G(X^p) for some

$$ c \in {{\mathbb{F}}}_{p} $$

and

$$ G(X) \in 1 + X{{\mathbb{F}}}_{p} {[}{[}X{]}{]} $$

, where

$$ E_{p}(X)=\hbox{exp}\left(\sum_{i=0}^{\infty} X^{p^{i}}/p^{i} \right) $$

is the Artin–Hasse exponential.