Some combinatorial properties of the Hurwitz series ring

TL;DR

This paper explores the algebraic structure of the Hurwitz series ring over a commutative ring, providing explicit formulas for invertible elements, examining sequence transforms, and introducing a new transform related to ultrametric spaces.

Contribution

It introduces a closed-form expression for invertible elements in the Hurwitz series ring using Bell polynomials and studies the automorphisms induced by Stirling transforms.

Findings

Explicit formula for invertible elements using Bell polynomials

Stirling transforms act as automorphisms of the Hurwitz series ring

A new sequence transform reveals an ultrametric dynamic space structure

Abstract

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.