Related papers: $A$-sequences, $Z$-sequence, and $B$-sequences of …
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 characterize certain Riordan arrays by their $A$-matrices and $\rho$ sequences. We conjecture the form of a generic $A$-matrix which leads to Somos $4$ sequences. We find an $A$-matrix that produces a Riordan quasi-involution, and we…
We introduce a method for describing Riordan matrices via recurrence relations along their diagonals. This provides a new structural description that complements the classical row-wise and column-wise constructions via the A-sequence. As an…
In this note, we explore links between Riordan arrays and lattice paths. We begin by describing Riordan arrays, and some of their generalizations, including rectifications and triangulations. We the consider Riordan array links to lattice…
Generalized Pascal matrix whose elements are generalized binomial coefficients is included in the group of generalized Riordan arrays. There is a special set of generalized Riordan arrays defined by parameter $q$. If $q=0$, they are…
This is the second paper of the paper series on multiple Riordan arrays. In this paper, based on the study of multiple Riordan arrays and the multiple Riordan group, we define multiple almost-Riordan arrays and find that the set of all…
We describe how the reversion of a series is related to convolutional recurrence relations for the series, and we place this relationship in the context of Riordan arrays. As an example of the approach, we give new recurrence relations for…
We present a parametric family of Riordan arrays which are obtained by multiplying any Riordan array with a generalized Pascal array. In particular, we focus on some interesting properties of one-parameter Catalan triangles. We obtain…
We define and characterize the $f$-matrices associated to Pascal-like matrices that are defined by ordinary and exponential Riordan arrays. These generalize the face matrices of simplices and hypercubes. Their generating functions can be…
We define and characterize the $\gamma$-matrix associated to Pascal-like matrices that are defined by ordinary and exponential Riordan arrays. We also define and characterize the $\gamma$-matrix of the reversions of these triangles, in the…
A vertical recursive relation approach to Riordan arrays is induced, while the horizontal recursive relation is represented by $A$- and $Z$-sequences. This vertical recursive approach gives a way to represent the entries of a Riordan array…
In this paper, we define Tribonacci and Tribonacci-Lucas matrix sequences and investigate their properties.
We consider Riordan arrays $\bigl(1/(1-t^{d+1}), ~ tp(t)\bigr)$. These are infinite lower triangular matrices determined by the formal power series $1/(1-t^{d+1})$ and a polynomial $p(t)$ of degree $d$. Columns of such matrix are eventually…
We study three operations on Riordan arrays. First, we investigate when the sum of Riordan arrays yields another Riordan array. We characterize the $A$- and $Z$-sequences of these sums of Riordan arrays, and also identify an analog for…
In this paper, we use the theory of Riordan matrices to introduce the notion of a Riordan graph. The Riordan graphs are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other…
We propose and investigate a bi-infinite matrix approach to the multiplication and composition of formal Laurent series. We generalize the concept of Riordan matrix to this bi-infinite context, obtaining matrices that are not necessarily…
For an integer $p\geq 2$ we construct vertical and horizontal one-pth Riordan arrays from a Riordan array. When $p=2$, one-pth Riordan arrays reduced to well known half Riordan arrays. The generating functions of the $A$-sequences of…
For a lower triangular matrix $(t_{n,k})$ we call the matrices with respective entries $(t_{2n-k,n})$ and $(t_{2n,n+k})$ the vertical and the horizontal halves. In this note, we discuss Riordan arrays whose halves are closely related to the…
This paper establishes new restrictions for attainable enhanced principal rank characteristic sequences (epr-sequences). These results are then used to classify two related families of sequences that are attainable by a real symmetric…
We study the Hankel transforms of sequences related to the central coefficients of a family of Pascal-like triangles. The mechanism of Riordan arrays is used to elucidate the structure of these transforms.