Hurwitz series as formal functions
Let \(A\) be a commutative ring with identity. The ring \(HA\) of Hurwitz series over \(A\) consists of countable sequences in \(A\) of the form \((a_0,a_1,a_2,\dots)\), or more simply \((a_n)\), where \(a_n\in A\) and \(n\in\mathbb N\). The ring operations are defined by \((a_n)+(b_n)=(a_n+b_n)\) and \((a_n)(b_n)=(c_n)\) where \(c_n=\sum_{k=0}^nC_k^na_kb_{n-k}\). The derivation \(\partial_A: HA\to HA\) is given by \(\partial_A((a_0,a_1,a_2,\dots))=(a_1,a_2,a_3,\dots)\). The authors continue the study, begun in [Commun. Algebra 25, 1845-1859 (1997; Zbl 0884.13013)], of properties of the ring \(HA\). In particular, they show that with its ``natural'' topology, \(HA\) is a complete metric space, and all of the natural mappings are continuous. There is a composition which satisfies the usual properties, i.e., it is continuous, associative, has a two-sided identity, is additive and multiplicative in the left (outer) factor, and satisfies the chain rule. It is also shown that \(HA\) provides formal solutions to homogeneous linear ordinary differential equations, using Picard's method of successive approximations. It is noteworthy that these results hold independently of the characteristic of the ring \(A\).
scientific article; zbMATH DE number 1420405