论文标题
Riordan阵列及其矩阵表示的垂直递归关系
The Vertical Recursive Relation of Riordan Arrays and Their Matrix Representation
论文作者
论文摘要
引起了Riordan阵列的垂直递归关系方法，而水平递归关系由$ a $ a - and $ z $ - 序列表示。这种垂直递归方法提供了一种方法，可以在$ g $的系数的递归线性组合中表示Riordan阵列$（g，f）$的条目。还给出了垂直递归关系的矩阵表示。所有这些矩阵的集合形成了一个称为准里族人组的组。通过使用Rook Triangle和Laguerre Triangle来定义水平递归关系的扩展和根据$ c $ - 和$ c $ - RIORDAN数组的扩展。这些扩展代表了研究某些三角矩阵条目的非线性递归关系的一种方法，这些矩阵来自Riordan阵列条目的线性递归关系。此外，Riordan阵列的垂直递归关系的矩阵表示提供了较低阶段和高阶有限Riordan阵列之间的转换，其中$ M $ th订单Riordan阵列由$（g，f）_m =（d_ {n，k}）_ {d_ {n，k}）此外，Riordan阵列的垂直关系方法为构建身份提供了统一的方法。
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 are defined with illustrations by using the rook triangle and the Laguerre triangle. Those extensions represent a way to study nonlinear recursive relations of the entries of some triangular matrices from linear recursive relations of the entries of Riordan arrays. In addition, the matrix representation of the vertical recursive relation of Riordan arrays provides transforms between lower order and high order finite Riordan arrays, where the $m$th order Riordan array is defined by $(g,f)_m=(d_{n,k})_{m\geq n,k\geq 0}$. Furthermore, the vertical relation approach to Riordan arrays provides a unified approach to construct identities.
