  We study some properties and perspectives of the Hurwitz series ring
$H_R[[t]]$, for a commutative ring with identity $R$. Specifically, we provide
a closed form for the invertible elements by means of the complete ordinary
Bell polynomials, we highlight some connections with well--known transforms of
sequences, and we see that the Stirling transforms are automorphisms of
$H_R[[t]]$. Moreover, we focus the attention on some special subgroups studying
their properties. Finally, we introduce a new transform of sequences that
allows to see one of this subgroup as an ultrametric dynamic space.
