Extensions of the Riordan Group

We consider two groups, F0= { g∈ F[ [ z] ] ∣ g(0 ) ≠ 0 } under multiplication, and F1= zF0 under composition where F is the real field R or complex field C. As observed in Sect. 3.3, 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 chapter, we develop the group of three-dimensional Riordan arrays [7] from an extension of the Riordan group R by F0. This concept extends to the group of multi-dimensional Riordan arrays. Moreover, we discuss the multivariate Riordan group [6, 17] defined by the semidirect product F0d⋊F1d for an integer d≥ 1. The two groups F0d and F1d are obtained, respectively, from F0 and F1 by extending the ring F[ [ z] ] of a single variable to the ring F[ [ z1, …, zd] ] of d variables. We will see some similarity with the Riordan group in a single variable, but its matrix representation, called a multivariate Riordan array, will be quite different to usual Riordan arrays.