Artin-Hasse exponential

In mathematics, the Artin-Hasse exponential is the power series given by

$E_p(x) = \exp(x + x^p/p + x^{p^2}/p^2 + x^{p^3}/p^3 +\cdots)$

[edit] Properties

The coefficients are p-integral; in other words, their denominators are not divisible by p. This follows from Dwork's lemma, which says that a power series f(x) = 1+... with rational coefficients has p-integral coefficients if and only if f(x^p)/f(x)^p ≡ 1 mod p.
The coefficient of xⁿ of n!E_p(x) is the number of elements of the symmetric group on n points of order a power of p. (This gives another proof that the coefficients are p-integral, using the fact that in a finite group of order divisible by d the number of elements of order dividing d is also divisible by d.)
It can be written as the infinite product