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

Elements of the Riordan group $\cal R$ 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 surprising 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 $\big(g(x), \, F(x)\big)$ has order $n$ in the group $\cal R$ actually needs to be checked. Using all this, our main result is to generalize C. Marshall [Congressus Numerantium, 229 (2017), 343-351] and prove: Given non-constant $g(x)$ satisfying necessary conditions, there exists a unique $F(x)$, given by an explicit formula, such that $\big(g(x), \, F(x)\big)$ is an involution in $\cal R$. Finally, as examples, we apply this theorem to ``aerated" series $h(x) = g(x^q),\ q\ \text{odd}$, to find the unique $K(x)$ such that $\big(h(x), K(x)\big)$ is an involution.