P分之一Riordan矩阵及恒等式构造

For an integer p≥2 we construct vertical and horizontal one-pth Riordan arrays from a Riordan array.When p=2 one-pth Riordan arrays are reduced to well known half Riordan arrays.The generating functions of the A-sequences of vertical and horizontal one-pth Riordan arrays are found.The vertical and horizontal one-pth Riordan arrays provide an approach to construct many identities.They can also be used to verify some well known identities readily.     

A symbolic treatment of Riordan arrays

We approach Riordan arrays and their generalizations via umbral symbolic methods. This new approach allows us to derive fundamental aspects of the theory of Riordan arrays as immediate consequences of the umbral version of the classical Abel?s identity for polynomials. In particular, we obtain a novel non-recursive formula for Riordan arrays and derive, from this new formula, some known recurrences and a new recurrence relation for Riordan arrays.     

Generalized Riordan arrays

In this paper, we generalize the concept of Riordan array. A generalized Riordan array with respect to c_n is an infinite, lower triangular array determined by the pair (g(t),f(t)) and has the generic element d_n,k=[tⁿ/c_n]g(t)(f(t))^k/c_k, where c_n is a fixed sequence of non-zero constants with c₀=1.We demonstrate that the generalized Riordan arrays have similar properties to those of the classical Riordan arrays. Based on the definition, the iteration matrices related to the Bell polynomials are special cases of the generalized Riordan arrays and the set of iteration matrices is a subgroup of the Riordan group. We also study the relationships between the generalized Riordan arrays and the Sheffer sequences and show that the Riordan group and the group of Sheffer sequences are isomorphic. From the Sheffer sequences, many special Riordan arrays are obtained. Additionally, we investigate the recurrence relations satisfied by the elements of the Riordan arrays. Based on one of the recurrences, some matrix factorizations satisfied by the Riordan arrays are presented. Finally, we give two applications of the Riordan arrays, including the inverse relations problem and the connection constants problem.     

Riordan 矩阵在广义 Motzkin 路计数中的应用

用Riordan矩阵的方法研究了具有4种步型的加权格路(广义Motzkin路)的计数问题,引入了一类新的计数矩阵,即广义Motzkin矩阵.同时给出了这类矩阵的Riordan表示,也得到了广义Motzkin路的计数公式.Catalan矩阵,Schrder矩阵和Motzkin矩阵都是广义Motzkin矩阵的特殊情形.     

An identity of Andrews and a new method for the Riordan array proof of combinatorial identities

We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G.E. Andrews. Several authors provided proofs of this identity, most of them rather involved or else relying on sophisticated number theoretical arguments. We present a new proof, quite simple and based on a Riordan array argument. The main point of the proof is the construction of a new Riordan array from a given Riordan array, by the elimination of elements. We extend the method and as an application we obtain other identities, some of which are new. An important feature of our construction is that it establishes a nice connection between the generating function of the A-sequence of a certain class of Riordan arrays and hypergeometric functions.     

Combinatorial sums and implicit Riordan arrays

In this paper we present the theory of implicit Riordan arrays, that is, Riordan arrays which require the application of the Lagrange Inversion Formula to be dealt with. We show several examples in which our approach gives explicit results, both in finding closed expressions for sums and, especially, in solving classes of combinatorial sum inversions.     

Proper generating trees and their internal path length

We find the generating function counting the total internal path length of any proper generating tree. This function is expressed in terms of the functions (d(t),h(t)) defining the associated proper Riordan array. This result is important in the theory of Riordan arrays and has several combinatorial interpretations.     

Riordan 矩阵的$(m,r,s)$-Halves

Riordan矩阵的垂直一半和水平一半已经被许多学者分别研究过.本文给出了Riordan矩阵的$(m,r,s)$-halves的定义.利用此定义能够统一的讨论Riordan矩阵的垂直一半和水平一半.作为应用,通过对Pascal和Delannoy矩阵的$(m,r,s)$-halves的研究,可以得到了一些与Fibonacci, Pell和Jacobsthal序列相关的等式.     

组合和的几个计算公式

In this paper, by Riordan array several computing formulas for the combinatorial sums are given.     

Ultrametrics, Banach’s fixed point theorem and the Riordan group

We interpret the reciprocation process in as a fixed point problem related to contractive functions for certain adequate ultrametric spaces. This allows us to give a dynamical interpretation of certain arithmetical triangles introduced herein. Later we recognize, as a special case of our construction, the so-called Riordan group which is a device used in combinatorics. In this manner we give a new and alternative way to construct the proper Riordan arrays. Our point of view allows us to give a natural metric on the Riordan group turning this group into a topological group. This construction allows us to recognize a countable descending chain of normal subgroups.     

Characterization of $(c)$-Riordan Arrays, Gegenbauer-Humbert-Type Polynomial Sequences, and $(c)$-Bell Polynomials

Here presented are the definitions of(c)-Riordan arrays and(c)-Bell polynomials which are extensions of the classical Riordan arrays and Bell polynomials.The characterization of(c)-Riordan arrays by means of the A-and Z-sequences is given,which corresponds to a horizontal construction of a(c)-Riordan array rather than its definition approach through column generating functions.There exists a one-to-one correspondence between GegenbauerHumbert-type polynomial sequences and the set of(c)-Riordan arrays,which generates the sequence characterization of Gegenbauer-Humbert-type polynomial sequences.The sequence characterization is applied to construct readily a(c)-Riordan array.In addition,subgrouping of(c)-Riordan arrays by using the characterizations is discussed.The(c)-Bell polynomials and its identities by means of convolution families are also studied.Finally,the characterization of(c)-Riordan arrays in terms of the convolution families and(c)-Bell polynomials is presented.     

Riordan阵与含有Genocchi数的一些恒等式

In this paper, using generating functions and Riordan arrays, we get some identities relating Genocchi numbers with Stirling numbers and Cauchy numbers.     

A generalization of Lucas polynomial sequence

In this paper, we obtain a generalized Lucas polynomial sequence from the lattice paths for the Delannoy numbers by allowing weights on the steps (1,0),(0,1) and (1,1). These weighted lattice paths lead us to a combinatorial interpretation for such a Lucas polynomial sequence. The concept of Riordan arrays is extensively used throughout this paper.     

Sequence characterization of Riordan arrays

In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the A- and Z-sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the A- and Z-sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. We also show how the sequence characterization is applied to construct easily a Riordan array. Finally, we give the characterizations relative to some subgroups of the Riordan group, in particular, of the hitting-time subgroup.     

Riordan群的反演链及在组合和中的应用

利用函数复合关系,本文在Ｒｉｏｒｄａｎ群中引入Ｒｉｏｒｄａｎ反演链的概念及其Ｒｉｏｒ－ｄａｎ反演链存在的充要条件,给出计算组合和式的递推方法．进一步讨论了二项式系数所对应的Ｒｉｏｒｄａｎ反演链问题,建立了一个Ｒｉｏｒｄａｎ求和公式,该式蕴含了某些与Ｆｉｂｏｎａｃｃｉ数相关的恒等式在内的一系列组合恒等式     

Riordan阵的差分性质及其应用

本文考虑了Riordan阵的差分性质, 并给出一些涉及经典组合序列的差分恒等式, 包括广义Stirling数, 第一类和第二类Stirling数, 第一类和第二类B型Stirling数以及Gegenbauer-Humbert型多项式.     

Identities on Bell polynomials and Sheffer sequences

In this paper, we study exponential partial Bell polynomials and Sheffer sequences. Two new characterizations of Sheffer sequences are presented, which indicate the relations between Sheffer sequences and Riordan arrays. Several general identities involving Bell polynomials and Sheffer sequences are established, which reduce to some elegant identities for associated sequences and cross sequences.     

Riordan arrays associated with Laurent series and generalized Sheffer-type groups

A relationship between a pair of Laurent series and Riordan arrays is formulated. In addition, a type of generalized Sheffer groups is defined by 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, and hitting-time subgroups of the groups are defined and discussed. A relationship between the generalized Sheffer groups with respect to different type of power series is presented. The equivalence of the defined Riordan array pairs and generalized Stirling number pairs is given. A type of inverse relations of various series is constructed by using pairs of Riordan arrays. Finally, several applications involving various arrays, polynomial sequences, special formulas and identities are also presented as illustrative examples.     

Inverse relations in Shapiro’s open questions

As an inverse relation, involution with an invariant sequence plays a key role in combinatorics and features prominently in some of Shapiro’s open questions (Shapiro, 2001). In this paper, invariant sequences are used to provide answers to some of these questions about the Fibonacci matrix and Riordan involutions.     

Hsu-Riordan 阵/partial monoid

本文首先对Shapiro的Riordan群进行了推广。给出了Hsu-Riordan partial monoid的概念，然后在此框架内，对徐利治先生的两类扩展型广义Stirling数偶进行了统一处理；建立了Hsu-Wang转换定理。Brown-Sprugnoli转换公式，以及广义Brown转换引理-它揭示了一些不同类型的Hsu-Riordan阵之间转换的方法。由此可产生大量的恒等式。