The n-th production matrix of a Riordan array

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 prod...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 703; pp. 63 - 77
Main Authors Ai, Hong-Zhang, Su, Xun-Tuan
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.12.2024
Subjects
Generating function
m-Catalan number
Production matrix
Riordan array
Total positivity
Generating function
Riordan array
05A19
15B48
15B36
m-Catalan number
05A15
Total positivity
Production matrix
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Summary: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.
ISSN:0024-3795
DOI:10.1016/j.laa.2024.08.022