One-pth Riordan Arrays in the Construction of Identities

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 esta…

The Vertical Recursive Relation of Riordan Arrays and Their Matrix Representation

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 $(g,f)$ in terms of a recursive linear combinations of the coefficients of $g$. A matrix representation of the vertical recursive relation is also given. The set of all those matrices forms a group, called the quasi-Riordan group. The extensions of the horizontal recursive relation and the vertical recursive relation in terms of $c$- and $C$- Riordan arrays …

Riordan arrays and applications via the classical umbral calculus

We use the classical umbral calculus to describe Riordan arrays. Here, a Riordan array is generated by a pair of umbrae, and this provides efficient proofs of several basic results of the theory such as the multiplication rule, the recursive properties, the fundamental theorem and the connection with Sheffer sequences. In particular, we show that the fundamental theorem turns out to be a reformulation of the umbral Abel identity. As an application, we give an elementary approach to the problem of extending integer powers of Riordan arrays to complex powers in such a way that additivity of the …

对称群特征标表中涉及Riordan数的新恒等式

New Identities in the Character Table of Symmetric Groups involving Riordan Numbers

针对Amdeberhan提出的对称群特征标表带符号列和等式，证明其在特殊情况下成立，得到不可约特征标度数和的新等式及Riordan数的新组合解释，是首个施加分拆奇偶性条件的相关定理。

关于Riordan阵列的和、导数与翻转

On Sums, Derivatives, and Flips of Riordan Arrays

该研究针对Riordan阵列开展三类运算研究，明确和运算对应阵列的序列特征、定义两种新运算并刻画其在子群上的作用结果，还生成了大量组合恒等式。

Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths

A partial Motzkin path is a path from $(0, 0)$ to $(n, k)$ in the $XOY$-plane that does not go below the $X$-axis and consists of up steps $U=(1, 1)$, down steps $D=(1, -1)$ and horizontal steps $H=(1, 0)$. A weighted partial Motzkin path is a partial Motzkin path with the weight assignment that all up steps and down steps are weighted by 1, the horizontal steps are endowed with a weight $x$ if they are lying on $X$-axis, and endowed with a weight $y$ if they are not lying on $X$-axis. Denote by $M_{n,k}(x, y)$ to be the weight function of all weighted partial Motzkin paths from $(0, 0)$ to $(…