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تسجيل مستخدم جديدElements of Finite Order in the Riordan Group and Their Eigenvectors
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Marshall M. Cohen
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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. Marshalls theorem, which finds the unique $F(x)$, given bi-invertible $g(x)$, such that $big(g(x), F(x))$ is an involution.
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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 Rior
dan 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.
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