In this note we determine all power series F(X) ∈ 1 + X double-struck F sign p[[X]] such that (F(X + Y))-1 F(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 ∈ double-struck F sign p and G(X) ∈ 1 + X double-struck F sign p[[X]] where Ep(X)= exp (∑i=0 ∞ Xp i/pi) is the Artin-Hasse exponential.
Mattarei, S. (2006). Exponential functions in prime characteristic. AEQUATIONES MATHEMATICAE, 71(3), 311-317 [10.1007/s00010-005-2816-4].
Exponential functions in prime characteristic
Mattarei, S
2006
Abstract
In this note we determine all power series F(X) ∈ 1 + X double-struck F sign p[[X]] such that (F(X + Y))-1 F(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 ∈ double-struck F sign p and G(X) ∈ 1 + X double-struck F sign p[[X]] where Ep(X)= exp (∑i=0 ∞ Xp i/pi) is the Artin-Hasse exponential.
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