n维Riordan阵列的相关性质

The Relevant Properties of n-Dimensional Riordan Arrays

摘要: $n$ 维Riordan阵列是一类广义的Riordan阵列。本文系统研究$n$ 维Riordan阵列的相关性质。首先针对三维Riordan阵列，探讨满足特定条件的元素分解问题。其次，利用广义巴纳赫不动点定理将三维Riordan阵列推广至高维，构造$n$ 维Riordan群上的双射，建立$n$ 维与$n-1$ 维Riordan群的关联。最后，通过该双射将对合性与伪对合性从$n-1$ 维推广到$n$ 维Riordan阵列，构造具有特定性质的$n$ 维Riordan群。

Abstract: An $n$ -dimensional Riordan array is a generalization of the classical Riordan array. This paper systematically investigates the properties of $n$ -dimensional Riordan arrays. First, for the 3-dimensional Riordan array, we discuss the element decomposition under certain conditions. Next, using the generalized Banach fixed-point theorem, we extend the 3-dimensional Riordan array to higher dimensions, construct a bijection on the $n$ -dimensional Riordan group, and establish a connection between the $n$ -dimensional and $\left(n-1\right)$ -dimensional Riordan groups. Finally, via this bijection, we extend the involution and pseudo-involution properties from $\left(n-1\right)$ -dimensional to $n$ -dimensional Riordan arrays, and construct an $n$ -dimensional Riordan group with specific properties.