Roots of Formal Power Series and New Theorems on Riordan Group Elements

Abstract

Elements of the Riordan group $\mathcal{R}$ over a field $\mathbb{F}$ of characteristic zero are infinite lower triangular matrices which are defined in terms of pairs of formal power series. We wish to bring to the forefront, as a tool in the theory of Riordan groups, the use of multiplicative roots $a(x{)}^{\frac{1}{n}}$ of elements $a(x)$ in the ring of formal power series over $\mathbb{F}$. Using roots, we give a Normal Form for non-constant formal power series, we prove a surprisingly simple Composition-Cancellation Theorem and apply this to show that, for a major class of Riordan elements (i.e., for non-constant $g(x)$ and appropriate $F(x)$), only one of the two basic conditions for checking that $(g(x),F(x))$ has order $n$ in the group $\mathcal{R}$ actually needs to be checked. Using all this, our main result is to generalize C. Marshall [6] and prove: Given non-constant $g(x)$ satisfying necessary conditions, there exists a unique $F(x)$, given by an explicit formula, such that $(g(x),F(x))$ is an involution in $\mathcal{R}$. Finally, as examples, we apply this theorem to “aerated” series $h(x)=g(x^r)$ to find the unique $K(x)$ such that $(h(x),K(x))$ is an involution.