Related papers: The Triple Riordan Group
The Riordan group is a set of infinite lower-triangular matrices defined by two generating functions, $g$ and $f$. The elements of the group are called Riordan arrays, denoted by $(g,f)$, and the $k$th column of a Riordan array is given by…
The classical Riordan groups associated to a given commutative ring are groups of infinite matrices (called Riordan arrays) associated to pairs of formal power series in one variable. The Fundamental Theorem of Riordan Arrays relates matrix…
We study a super group of the group of Riordan arrays, where the elements of the group are given by a triple of power series. We show that certain subsets are subgroups, and we identify a normal subgroup whose cosets correspond to Riordan…
The generalized Riordan group consists of infinite lower triangular matrices that correspond to certain operators in the space of formal power series. Each such group contains the matrix (generalized Pascal matrix), elements of which are…
We show how to define, for every Riordan group element $(g(x), f(x))$, an involution in the Riordan group. More generally, we show that for every pseudo-involution $P$ in the Riordan group, we can define a new involution beginning with an…
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…
The Riordan group, along with its constituent elements, Riordan arrays, has been a tool for combinatorial exploration since its inception in 1991. More recently, this group has made an appearance in the area of mathematical physics, where…
This is the first paper of a sequence papers on the multiple Riordan group and the multiple Riordon type arrays. We give a comprehensive discussion of the multiple Riordan arrays and characterize them by an $A$-sequence and multiple…
We define a group of lower-triangular matrices whose columns are defined by power series. This group can be seen as a generalization of the (ordinary) Riordan group and the double Riordan group. Elements of this group are defined by three…
Riordan matrices are infinite lower triangular matrices that correspond to certain operators in the space of formal power series. In this paper, we introduce similar matrices for the space of formal Dirichlet series. We show that these…
We calculate the derived series of the Riordan group. To do that, we study a nested sequence of its subgroups, herein denoted by $\mathcal G_k$. By means of this sequence, we first obtain the n-th commutator subgroup of the Associated…
Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty finite product of conjugates of $g$ equals to the identity, then $g$ is called a generalized torsion element. The minimum number of conjugates in such a product is…
The aim of this paper is to describe the definitions and main properties of three generalizations of the group concept, namely: groupoid, generalized group and almost groupoid. Some constructions of these algebraic structures and…
We provide an alternative description of the group of Riordan arrays, by using two power series of the form $\sum_{n=0}^{\infty} g_n x^n$, where $g_0 \ne 0$ to build a typical element of the constructed group. We relate these elements to…
The Riordan group is the semi-direct product of a multiplicative group of invertible series and a group, under substitution, of non units. The Riordan near algebra, as introduced in this paper, is the Cartesian product of the algebra of…
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…
The Riordan group ${\cal R}$ over the field ${\mathbb F}_2$ is a split extension of the Appell subgroup by the Nottingham group ${\cal N}({\mathbb F}_2)$. Using the lower central series of the Nottingham group obtained by C. Leedham-Green…
It is well-known that pythagorean triples can be represented by points of the unit circle with rational coordinates. These points form an abelian group, and we describe its structure. This structural description yields, almost immediately,…
We prove that any element in the group generated by the Riordan involutions is the product of at most four of them. We also give a description of this subgroup as a semidirect product of a special subgroup of the commutator subgroup and the…
In this paper, we discuss centralizers in the Riordan group. We will see that Fa\`a di Bruno's formula is an application of the Fundamental Theorem of Riordan arrays. Then the composition group of formal power series in ${\cal F}_1$ is…