Riordan Arrays Research Articles

Riordan Array Approach to the Coefficients of Ramanujan’s Harmonic Number Expansion

In this paper, by the Riordan array method, we show that in Ramanujan’s asymptotic expansion of the harmonic numbers, the coefficients given by Villarino satisfy the recurrence given by Chen and Cheng. This gives an affirmative answer to Chen and Cheng’s open problem.

Arithmetic into geometric progressions through Riordan arrays

In this paper, by using Riordan arrays and a particular model of lattice paths, we are able to generalize in several ways an identity proposed by Lou Shapiro by giving both an algebraic and a combinatorial proof. The identities studied in this paper allow us to move from an arithmetic progression, and other C-finite sequences, to a geometric progression in terms of Riordan array transformations and vice versa, via the Riordan array inverse.

Riordan arrays and generalized Lagrange series

The theory of Riordan arrays studies the properties of formal power series and their sequences. The notion of generalized Lagrange series proposed in the present paper is intended to fill the gap in the methodology of this theory. Generalized Lagrange series appear in it implicitly, as various equalities. No special notation is provided for these series, although particular cases of these series are generalized binomial and generalized exponential series. We give the definition of generalized Lagrange series and study their relationship with ordinary Riordan arrays and, separately, with Riordan exponential arrays.

Applications of Riordan matrix functions to Bernoulli and Euler polynomials

We define Riordan matrix functions associated with Riordan arrays and study their algebraic properties. We also give their applications in the construction of new classes of Bernoulli and Euler polynomials and Bernoulli and Euler numbers, referred to as the duals and conjugate Bernoulli and Euler polynomials and dual and conjugate Bernoulli and Euler numbers, respectively.

Row sums and alternating sums of Riordan arrays

Here we use row sum generating functions and alternating sum generating functions to characterize Riordan arrays and subgroups of the Riordan group. Numerous applications and examples are presented which include the construction of Girard–Waring type identities. We also show the extensions to weighted sum (generating) functions, called the expected value (generating) functions of Riordan arrays.

Combinatorics of a generalized Narayana identity

The Narayana identity is a well-known formula that expresses the classical Catalan numbers as sums of the ordinary Narayana numbers. In this paper we generalize the Narayana identity to a family of Riordan arrays including the array of ballot numbers, the classical Catalan triangle and several generalized Catalan triangles recently studied. A combinatorial description based on non-crossing partitions is given for this identity, for the column-recursive rule, and for the Sheffer sequence associated with any array of the family.

Shift operators defined in the Riordan group and their applications

In this paper, we discuss a linear operator T defined in Riordan group R by using the upper shift matrix U and lower shift matrix UT, namely for each R∈R, T:R↦URUT. Some isomorphic properties of the operator T and the structures of its range sets for different domains are studied. By using the operator T and the properties of Bell subgroup of R, the Riordan type Chu–Vandermonde identities and the Riordan equivalent identities of Format Last Theorem and Beal Conjecture are shown. The applications of the shift operators to the complementary Riordan arrays and to the Riordan involutions and Riordan pseudo-involutions are also presented.

Some identities of the r-Whitney numbers

In this paper we establish some algebraic properties involving r-Whitney numbers and other special numbers, which generalize various known identities. These formulas are deduced from Riordan arrays. Additionally, we introduce a generalization of the Eulerian numbers, called r-Whitney–Eulerian numbers and we show how to reduce some infinite summation to a finite one.

Generalized Riordan groups and operators on polynomials

We present an approach to generalized Riordan arrays which is based on operations in one large group of lower triangular matrices. This allows for direct proofs of many properties of weighted Sheffer sequences, and shows that all the groups arising from different weights are isomorphic since they are conjugate. We also prove a result about the intersection of two generalized Riordan groups with different weights.

Riordan Arrays and Generalized Lagrange Series

Теория матриц Риордана изучает свойства формальных степенных рядов и их последовательностей. Понятие обобщенных рядов Лагранжа, предлагаемое в данной статье, призвано заполнить пробел в методологии этой теории. Обобщенные ряды Лагранжа присутствуют в ней неявно, в виде различных равенств. Специального обозначения они не имеют, хотя частными случаями этих рядов являются обобщенный биномиальный и обобщенный экспоненциальный ряд. В статье дается определение обобщенных рядов Лагранжа, рассматривается их связь с обыкновенными матрицами Риордана и отдельно с экспоненциальными матрицами Риордана. Библиография: 11 названий.

$q$-Riordan array for $q$-Pascal matrix and its inverse matrix

In this paper, we prove the $q$-analogue of the fundamental theorem of Riordan arrays. In particular, by defining two new binary operations $\ast_{q} $ and $\ast _{1/q}$, we obtain a $q$-analogue of the Riordan representation of the $q$-Pascal matrix. In addition, by aid of the $q$-Lagrange expansion formula we get $q$-Riordan representation for its inverse matrix.

Riordan arrays, generalized Narayana triangles, and series reversion

Using elements of the group of Riordan arrays we define a family of generalized Narayana triangles and their associated generalized Catalan numbers, and study their links to series reversion. In particular we use Lagrange inversion techniques to determine the generating functions for these generalized Catalan numbers.

Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups

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 prefunctions). Thereby, various properties are studied arising in Riordan arrays, in the Riordan group and, more specifically, in the "striped" Riordan subgroups; further, a striped quasigroup and a semigroup are also examined. A few applications to combinatorial structures are also briefly addressed in the Appendix.

Strong [formula omitted]-log-convexity of the Eulerian polynomials of Coxeter groups

In this paper we prove the strong q-log-convexity of the Eulerian polynomials of Coxeter groups using their exponential generating functions. Our proof is based on the theory of exponential Riordan arrays and a criterion for determining the strong q-log-convexity of polynomial sequences, whose generating functions can be given by a continued fraction. As applications, we get the strong q-log-convexity of the Eulerian polynomials of types An,Bn, their q-analogue and the generalized Eulerian polynomials associated to the arithmetic progression {a,a+d,a+2d,a+3d,…} in a unified manner.

A Generalization of the $k$-Bonacci Sequence from Riordan Arrays

In this article, we introduce a family of weighted lattice paths, whose step set is $\{H=(1,0), V=(0,1), D_1=(1,1), \dots, D_{m-1}=(1,m-1)\}$. Using these lattice paths, we define a family of Riordan arrays whose sum on the rising diagonal is the $k$-bonacci sequence. This construction generalizes the Pascal and Delannoy Riordan arrays, whose sum on the rising diagonal is the Fibonacci and tribonacci sequence, respectively. From this family of Riordan arrays we introduce a generalized $k$-bonacci polynomial sequence, and we give a lattice path combinatorial interpretation of these polynomials. In particular, we find a combinatorial interpretation of tribonacci and tribonacci-Lucas polynomials.

On the r-Central Coefficient Matrices of the Catalan Triangles

We introduce the definition of the r-central coefficient matrices of a given Riordan array. Applying this definition and Lagrange Inversion Formula, we can calculate the r-central coefficient matrices of Catalan triangles and obtain some interesting triangles and sequences.

Sums of Involving the Harmonic Numbers and the Binomial Coefficients

Let the numbers be defined by , where and are the exponential complete Bell polynomials. In this paper, by means of the methods of Riordan arrays, we establish general identities involving the numbers , binomial coefficients and inverse of binomial coefficients. From these identities, we deduce some identities involving binomial coefficients, Harmonic numbers and the Euler sum identities. Furthermore, we obtain the asymptotic values of some summations associated with the numbers by Darboux’s method.

Total positivity of Riordan arrays

We present sufficient conditions for the total positivity of Riordan arrays. As applications we show that many well-known combinatorial triangles are totally positive and many famous combinatorial numbers are log-convex in a unified approach.

Summations of Inverse Generalized Harmonic Numbers with Riordan Arrays

By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we using connections between the Stirling numbers of both kinds and other inverse generalized harmonic numbers. we proved some combinatorial sums and inverse generalized harmonic number identities.

Diagonal Asymptotics for Products of Combinatorial Classes

We generalize and improve recent results by Bóna and Knopfmacher and by Banderier and Hitcz-enko concerning the joint distribution of the sum and number of parts in tuples of restricted compositions. Specifically, we generalize the problem to general combinatorial classes and relax the requirement that the sizes of the compositions be equal. We extend the main explicit results to enumeration problems whose counting sequences are Riordan arrays. In this framework, we give an alternative method for computing asymptotics in the supercritical case, which avoids explicit diagonal extraction.