Generalized Riordan arrays
Generalized Riordan arrays
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DOI：
10.1016/j.disc.2007.12.037
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发表时间：
2008-12
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影响因子：
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通讯作者：
Weiping Wang;Tian-ming Wang
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文献类型：
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作者：
Weiping Wang;Tian-ming Wang
本文推广了Riordan阵列的概念。关于Cn的广义Riordan阵列是由(g(T)，f(T))对确定的无限下三角阵列，具有一般元素Dn，k=[tn/Cn]g(T)(f(T))k/Ck，其中Cn是一个固定的非零常数序列，c0=1。我们证明了广义Riordan阵列具有与经典Riordan阵列相似的性质。基于该定义，与Bell多项式相关的迭代矩阵是广义Riordan阵列的特例，迭代矩阵的集合是Riordan群的子群。我们还研究了广义Riordan阵列和Sheffer序列之间的关系，证明了Riordan群和Sheffer序列群是同构的。由Sheffer序列得到了许多特殊的Riordan阵列。此外，我们还研究了Riordan阵列元素所满足的递推关系。基于其中一个递归，给出了Riordan阵列所满足的一些矩阵分解。最后，我们给出了Riordan阵列的两个应用，包括反关系问题和连接常数问题。
In this paper, we generalize the concept of Riordan array. A generalized Riordan array with respect to cnis an infinite, lower triangular array determined by the pair (g(t),f(t)) and has the generic element dn,k=[tn/cn]g(t)(f(t))k/ck, where cnis 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.