On the Artin-Hasse exponential series

Abstract

1. Professor G. Whaples has kindly drawn my attention to the very similar properties enjoyed by the series which I called the Witt hyperexponential in a recent paper [2], and a series which he had previously defined, using the Artin-Hasse exponential series [5]; the main fact is that both series define a homomorphism of the Witt group W onto the multiplicative group W*. In answer to his questions, I propose in this note to clear up completely that relationship, by determining all formal power series which define such homomorphisms, in other words, what one might call the formal characters of the group W; it turns out that the Artin-Hasse-Whaples series is the simplest member of that family, from which all others can be deduced by a simple transformation. I am indebted to Professor Whaples for several useful remarks and comments, as well as for pointing out a slight error in one of my original proofs.