{"title":"The n-th production matrix of a Riordan array","authors":"Hong-Zhang Ai, Xun-Tuan Su","doi":"10.1016/j.laa.2024.08.022","DOIUrl":null,"url":null,"abstract":"<div><p>The production matrix plays an important role in characterizing a Riordan array. Recently, Barry explored the notion of the <em>n</em>-th production matrix and characterized the Riordan arrays corresponding to the second and third production matrices respectively. This paper is devoted to study the <em>n</em>-th production matrix and its corresponding Riordan arrays systematically. Our work is threefold. First, we show that every <em>n</em>-th production matrix can be factorized into a product of <em>n</em> matrices associated with the ordinary production matrix. Second, we prove a characterization of the Riordan array corresponding to the <em>n</em>-th production matrix, which was conjectured by Barry. Third, we claim that if the ordinary production matrix of a Riordan array is totally positive, so are the <em>n</em>-th production matrix and its corresponding Riordan arrays. Our results are illustrated by the generalized Catalan array which includes many well-known Riordan arrays as special cases.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524003562","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The production matrix plays an important role in characterizing a Riordan array. Recently, Barry explored the notion of the n-th production matrix and characterized the Riordan arrays corresponding to the second and third production matrices respectively. This paper is devoted to study the n-th production matrix and its corresponding Riordan arrays systematically. Our work is threefold. First, we show that every n-th production matrix can be factorized into a product of n matrices associated with the ordinary production matrix. Second, we prove a characterization of the Riordan array corresponding to the n-th production matrix, which was conjectured by Barry. Third, we claim that if the ordinary production matrix of a Riordan array is totally positive, so are the n-th production matrix and its corresponding Riordan arrays. Our results are illustrated by the generalized Catalan array which includes many well-known Riordan arrays as special cases.
生成矩阵在描述瑞奥德恩数组的特征时起着重要作用。最近，巴里探索了 nth 生产矩阵的概念，并分别描述了与第二和第三生产矩阵相对应的瑞尔丹数组。本文致力于系统研究 nth 生产矩阵及其对应的瑞尔丹数组。我们的工作包括三个方面。首先，我们证明了每个 n 次生产矩阵都可以因式分解为与普通生产矩阵相关的 n 个矩阵的乘积。其次，我们证明了与第 n 个生产矩阵相对应的瑞尔丹数组的特征，这是巴里的猜想。第三，我们声称，如果一个瑞尔丹数组的普通生产矩阵是全正的，那么第 n 个生产矩阵及其对应的瑞尔丹数组也是全正的。我们的结果通过广义加泰罗尼亚数组来说明，其中包括许多著名的瑞尔丹数组特例。
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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