相关论文: Pseudo-involutions in the Riordan group and Chebys…
We consider pseudo-involutions in the Riordan group where the generating function $g$ for the first column of a Riordan array satisfies a palindromic or near-palindromic functional equation. For those types of equations, we find, for very…
Each numerical sequence $\left( {{b}_{0}},{{b}_{1}},{{b}_{2}},... \right)$ with the generating function $B\left( x \right)$ defines the pseudo-involution in the Riordan group $\left( 1,xg\left( x \right) \right)$ such that $g\left( x…
Riordan arrays, denoted by pairs of generating functions (g(z), f(z)), are infinite lower-triangular matrices that are used as combinatorial tools. In this paper, we present Riordan and stochastic Riordan arrays that have connections to the…
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 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…
Let $\left( g\left( x \right),xg\left( x \right) \right)$ be a Riordan matrix from the Bell subgroup. We denote ${{\left( g\left( x \right),xg\left( x \right) \right)}^{\varphi }}=\left( {{g}^{\left( \varphi \right)}}\left( x…
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…
We show that the coefficient array of a family of Chebyshev moments defines an involution in the group of Riordan arrays. We then extend this result to certain families of $d$-orthogonal polynomials.
Chebyshev polynomials and their modifications are attributes of various fields of mathematics. In particular, they are generating functions of the rows elements of certain Riordan matrices. In paper, we give a selection of some…
In this article, the 2-iterated Sheffer polynomials are introduced by means of generating function and operational representation. Using the theory of Riordan arrays and relations between the Sheffer sequences and Riordan arrays, a…
We show that certain Riordan arrays have generating functions that can be expressed as continued fractions of Jacobi and Thron type. We investigate the inverses of such arrays, which in certain circumstances can also have generating…
Sigmoid functions play an important role in many areas of applied mathematics, including machine learning, population dynamics and probability. We place the study of sigmoid functions in the context of the derivative sub-group of the group…
Several authors have examined connections among restricted permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for involutions which avoid 3412. Our results…
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…
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…
The main object of the paper is to reveal connections between Chebyshev polynomials of the first and second kinds and Fibonacci polynomials introduced by Catalan. This is achieved by relating the respective (ordinary and exponential)…
We define a three parameter family of Bell pseudo-involutions in the Riordan group. The defining sequences have generating functions that are expressible as continued fractions. We indicate that the Hankel transforms of the defining…
We study a one-parameter family of binomial-convolution operators acting on sequences. These operators form an additive semigroup with an explicit inverse, and they subsume iterated classical binomial transforms as a special case. We…
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…
The connection between the generating functions of various sets of tableaux and the appropriate families of quasisymmetric functions is a significant tool to give a direct analytical proof of some advanced bijective results and provide new…