The group of multi-dimensional Riordan arrays

We first consider two groups, F0={g∈C[[z]]|g(0)≠0} under multiplication and F1=zF0 under composition, where C[[z]] is the ring of formal power series over the complex field. It is known that the Riordan group R is isomorphic to the semidirect product F0⋊F1. It may be viewed as a group extension of F1 by F0. In this paper, the group of three-dimensional Riordan arrays is obtained from an extension of the group R by F0. This concept extends to the group of multi-dimensional Riordan arrays. As an application, we illustrate the use of the three-dimensional Riordan array in multiple combinatorial sums.