Elements of Finite Order in the Riordan Group

Marshall M. Cohen

Published 2018-06-17Version 1

We consider elements $\big(g(x), F(x)\big)$ of finite order in the Riordan group over a field $\cal F$ of characteristic $0$. We solve for all integers $n\geq 2$ the two fundamental questions posed by L. Shapiro (2001) for the case $n = 2$ ("involutions"). Given a formal power series $F(x) = \omega x + F_2x^2 + \cdots$, and an integer $n\geq 2$, Theorem 1 states exactly which $g(x)$ make $\big(g(x), F(x))$ a Riordan element of order $n$. Theorem 2 classifies finite-order Riordan group elements up to conjugation. We then relate our work to papers (2008, 2013) of Cheon and Kim which motivated this paper. We supply a missing proof in the first paper and we solve the Open question in Section 2 of the second paper. Finally we show how this circle of ideas gives a new proof of C. Marshall's theorem (2017), which finds the unique $F(x)$, given bi-invertible $g(x)$, such that $\big(g(x), F(x))$ is an involution.