An alternative presentation of the m-tuple Riordan group

We provide an embedding of the m-tuple Riordan group, introduced by Davenport, Shapiro, and Woodson, into the direct product Rm of m copies of the Riordan group R. Within this framework, we define a subgroup R[m]⊂Rm, whose elements can be conveniently represented by arrays [f0,…,fm] of formal power series with nonzero constant term. This construction allows for straightforward manipulation of m-tuple Riordan arrays and provides a natural extension of the classical Riordan group theory to the m-tuple Riordan setting.