On the operations of sequences in rings and binomial type sequences

TL;DR

This paper explores algebraic operations on sequences over rings, revealing new relations, automorphisms, and a novel isomorphism that aids in characterizing binomial type sequences.

Contribution

It introduces a new isomorphism between sequence structures and develops methods for characterizing binomial type sequences.

Findings

Identifies relations between binomial convolution and composition operations.

Highlights automorphisms including the Stirling transform as special cases.

Provides a new method for generating binomial type sequences.

Abstract

Given a commutative ring with identity $R$, many different and interesting operations can be defined over the set $H_R$ of sequences of elements in $R$. These operations can also give $H_R$ the structure of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well--known transforms of sequences (such as the Stirling transform) are special cases of these automorphisms. Moreover, we introduce a novel isomorphism between $H_R$ equipped with the componentwise sum and the set of the sequences starting with 1 equipped with the binomial convolution product. Finally, thanks to this isomorphism, we find a new method for characterizing and generating all the binomial type sequences.