Riordan Groups in higher Dimensions

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.