Abstract

In this note we determine all power series

F(X)∈1+XF p [[X]]
F(X)∈1+XFp[[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) = Ep(cX)· G(Xp) for some

c∈F p
c∈Fp
and

G(X)∈1+XF p [[X]]
G(X)∈1+XFp[[X]]
, where

E p (X)=exp(∑ i=0 ∞ X p i /p i )
Ep(X)=exp(∑i=0∞Xpi/pi)
is the Artin–Hasse exponential.