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

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.