Finding an isomorphism between the Riordan group and a subgroup of the double Riordan group

The Riordan group R is a set of infinite lower-triangular matrices defined by two generating functions, g and f .The elements of R are called Riordan arrays and denoted by (g, f ), where f is the multiplier function of the array.The 0 th column of a Riordan array is defined by g, and the k th column is given by the generating function gf k .Similarly, the Double Riordan group DR is a set of infinite lower-triangular matrices defined by three generating functions, g, f 1 , and f 2 , where g is an even function and the multiplier functions f 1 and f 2 are odd.Since the introduction of the Double Riordan group by Davenport, Shapiro, and Woodson in 2011, an open question remained as to whether there is a subgroup of the Double Riordan group that is isomorphic to the Riordan group.This paper introduces monomorphism theorems that answer this question and extend the results to the k-Riordan group.We also use these theorems to define elements of the Double Riordan group that are analogous to pseudo-involutions of the Riordan group.