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Register a new userRoots of Formal Power Series and New Theorems on Riordan Group Elements
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Authors
Marshall M. Cohen
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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 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 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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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.
Fixing a positive integer $r$ and $0 le k le r-1$, define $f^{langle r,k rangle}$ for every formal power series $f$ as $ f(x) = f^{langle r,0 rangle}(x^r)+xf^{langle r,1 rangle}(x^r)+ cdots +x^{r-1}f^{langle r,r-1 rangle}(x^r).$ Jochemko recently showed that the polynomial $U^{n}_{r,k}, h(x) := left( (1+x+cdots+x^{r-1})^{n} h(x) right)^{langle r,k rangle}$ has only nonpositive zeros for any $r ge deg h(x) -k$ and any positive integer $n$. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial $h(x)$ of a lattice polytope of dimension $n$, which states that $U^{n}_{r,0},h(x)$ has only negative, real zeros whenever $rge n$. In this paper, we provide an alternative approach to Beck and Stapledons conjecture by proving the following general result: if the polynomial sequence $left( h^{langle r,r-i rangle}(x)right)_{1le i le r}$ is interlacing, so is $left( U^{n}_{r,r-i}, h(x) right)_{1le i le r}$. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontais result on the interlacing property of some refinements of the descent generating functions for colored permutations. Besides, we derive a Carlitz identity for refined colored permutations.
The group of almost Riordan arrays contains the group of Riordan arrays as a subgroup. In this note, we exhibit examples of pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays.
We continue the first and second authors study of $q$-commutative power series rings $R=k_q[[x_1,ldots,x_n]]$ and Laurent series rings $L=k_q[[x^{pm 1}_1,ldots,x^{pm 1}_n]]$, specializing to the case in which the commutation parameters $q_{ij}$ are all roots of unity. In this setting, $R$ is a PI algebra, and we can apply results of De Concini, Kac, and Procesi to show that $L$ is an Azumaya algebra whose degree can be inferred from the $q_{ij}$. Our main result establishes an exact criterion (dependent on the $q_{ij}$) for determining when the centers of $L$ and $R$ are commutative Laurent series and commutative power series rings, respectively. In the event this criterion is satisfied, it follows that $L$ is a unique factorization ring in the sense of Chatters and Jordan, and it further follows, by results of Dumas, Launois, Lenagan, and Rigal, that $R$ is a unique factorization ring. We thus produce new examples of complete, local, noetherian, noncommutative, unique factorization rings (that are PI domains).
Fields of generalised power series (or Hahn fields), with coefficients in a field and exponents in a divisible ordered abelian group, are a fundamental tool in the study of valued and ordered fields and asymptotic expansions. The subring of the series with non-positive exponents appear naturally when discussing exponentiation, as done in transseries, or integer parts. A notable example is the ring of omnific integers inside the field of Conways surreal numbers. In general, the elements of such subrings do not have factorisations into irreducibles. In the context of omnific integers, Conway conjectured in 1976 that certain series are irreducible (proved by Berarducci in 2000), and that any two factorisations of a given series share a common refinement. Here we prove a factorisation theorem for the ring of series with non-positive real exponents: every series is shown to be a product of irreducible series with infinite support and a factor with finite support which is unique up to constants. From this, we shall deduce a general factorisation theorem for series with exponents in an arbitrary divisible ordered abelian group, including omnific integers as a special case. We also obtain new irreducibility and primality criteria. To obtain the result, we prove that a new ordinal-valued function, which we call degree, is a valuation on the ring of generalised power series with real exponents, and we formulate some structure results on the associated RV monoid.
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