Riordan Groups in higher Dimensions

TL;DR

This paper extends the classical Riordan groups to higher dimensions involving multiple variables, establishing a fundamental theorem that relates matrix multiplication to group actions on multivariate formal power series.

Contribution

It introduces higher-dimensional Riordan groups with formal power series in several variables and proves an analogue of the Fundamental Theorem for these groups.

Findings

Defined higher-dimensional Riordan groups.

Established the fundamental theorem in multivariate context.

Discussed related Laurent series groups and posed open questions.

Abstract

The classical Riordan groups associated to a given commutative ring are groups of infinite matrices (called Riordan arrays) associated to pairs of formal power series in one variable. The Fundamental Theorem of Riordan Arrays relates matrix multiplication to two group actions on such series, namely formal (convolution) multiplication and formal composition. We define the analogous Riordan groups involving formal power series in several variables, and establish the analogue of the Fundamental Theorem in that context. We discuss related groups of Laurent series and pose some questions.