A relationship between a pair of Laurent series and Riordan arrays is formulated. In addition, a type of generalized Sheffer groups is defined using Riordan arrays with respect to power series with non-zero coefficients. The isomorphism between a generalized Sheffer group and the group of the Riordan arrays associated with Laurent series is established. Furthermore, Appell, associated, Bell, an...

In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg–Weyl, its Bargmann–Fock representation with differential operators and the associated one-parameter group. Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e., substitutions with prefu...

In this short note we focus on self-inverse Sheffer sequences and involutions in the Riordan group. We translate the results of Brown and Kuczma on self-inverse sequences of Sheffer polynomials to describe all involutions in the Riordan group.

The purpose of this paper is twofold. As the first goal, we show that three different classes of random walks are counted by the Pell numbers. The calculations are done using a convenient technique that involves the Riordan group. This leads to the second goal, which is to demonstrate this convenient technique. We also construct bijections among Pell, certain Motzkin, and certain Schröder walks...

We study integer sequences and transforms that operate on them. Many of these transforms are defined by triangular arrays of integers, with a particular focus on Riordan arrays and Pascal-like arrays. In order to explore the structure of these transforms, use is made of methods coming from the theory of continued fractions, hypergeometric functions, orthogonal polynomials and most importantly f...

We use the Lagrange-Bürmann inversion theorem to characterize the generating function of the central coefficients of the elements of the Riordan group of matrices. We apply this result to calculate the generating function of the central elements of a number of explicit Riordan arrays, defined by rational expressions, and in two cases we use the generating functions thus found to calculate the H...

2 The “algebra” K[[x]] ⋊ M of formal power series under multiplication and substitution 3 2.1 Basics on formal power series . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 “Algebra” of formal power series under substitution . . . . . . . . . . . . 4 2.2.1 Right-distributive algebras . . . . . . . . . . . . . . . . . . . . . . 4 2.2.2 Substitution of formal power series . . . . . . . . . ....