Exponential functions in prime characteristic

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.