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- W2807844523 abstract "We consider elements of finite order in the Riordan group $cal R$ over a field of characteristic $0$. Viewing $cal R$ as a semi-direct product of groups of formal power series, we solve, for all $n geq 2$, two foundational questions posed by L. Shapiro for the case $n = 2$ (`involutions'): Given a formal power series $F(x)$ of finite compositional order and an integer $ngeq 2$, Theorem 1 states, exactly which $g(x)$ make $big(g(x), F(x)big)$ a Riordan element of order $n$. Theorem 2 classifies finite-order Riordan group elements up to conjugation in $cal R$. Viewing $cal R$ as a group of infinite lower triangular matrices, we interpret Theorem 1 in terms of existence of eigenvectors and Theorem 2 as a normal form for finite order Riordan arrays under similarity. These lead to Theorem 3, a formula for all eigenvectors of finite order Riordan arrays; and we show how this can lead to interesting combinatorial identities. We then relate our work to papers of Cheon and Kim which motivated this paper and we solve the Open question which they posed. Finally, this circle of ideas gives a new proof of C. Marshall's theorem, which finds the unique $F(x)$, given bi-invertible $g(x)$, such that $big(g(x), F(x))$ is an involution." @default.
- W2807844523 created "2018-06-21" @default.
- W2807844523 creator A5044880753 @default.
- W2807844523 date "2018-06-17" @default.
- W2807844523 modified "2023-09-27" @default.
- W2807844523 title "Elements of Finite Order in the Riordan Group" @default.
- W2807844523 hasPublicationYear "2018" @default.
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