Elements of finite order in the Riordan group and their eigenvectors

Abstract

We consider elements of finite order in the Riordan group R over a field F of characteristic 0. We solve, for all n≥2, two foundational questions posed by L. Shapiro (2001) for the case n=2 (“involutions”): Given F(x) of finite compositional order n, Theorem 1 gives a formula for g(x) showing that those g(x) which make (g(x),F(x))∈R a Riordan element of order n are precisely those for which the matrix (g(x),F(x)) has an eigenvector (h0,h1,…,)T with h0≠0. Theorem 2 classifies finite-order Riordan group elements up to conjugation in R and gives a normal form for finite order Riordan arrays under similarity. This leads to Theorem 3, a formula for all eigenvectors of finite order Riordan arrays, and results in interesting combinatorial identities. We relate our work to 2008 and 2013 papers of Cheon and Kim which motivated this paper; and we solve their Open question. This circle of ideas closes with a new proof of C. Marshall's theorem, which finds the unique F(x), given bi-invertible g(x), such that (g(x),F(x)) is an involution.