Total Positivity of Almost-Riordan Arrays Total Positivity of Almost-Riordan Arrays

In this paper we study the total positivity of almost-Riordan arrays ( d ( t ) | g ( t ) , f ( t ) ) and establish its necessary conditions and sufficient conditions, particularly, for some well-used formal power series d ( t ). We present a semidirect product of an almost-array and use it to transf...

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Vydané v:	Graphs and combinatorics Ročník 41; číslo 6; s. 115
Hlavní autori:	He, Tian-Xiao, Słowik, Roksana
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Tokyo Springer Japan 01.12.2025 Springer Nature B.V
Predmet:	Arrays Combinatorics Engineering Design Mathematics Mathematics and Statistics Original Paper Power series Almost-Riordan array Almost-Riordan group 11B83 Riordan array 15A06 05A19 15B36 05A05 05A15 Riordan group Quasi-Riordan array
ISSN:	0911-0119, 1435-5914
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Abstract	In this paper we study the total positivity of almost-Riordan arrays ( d ( t ) \| g ( t ) , f ( t ) ) and establish its necessary conditions and sufficient conditions, particularly, for some well-used formal power series d ( t ). We present a semidirect product of an almost-array and use it to transfer a total positivity problem for an almost-Riordan array to the total positivity problem for a quasi-Riordan array. We find the sequence characterization of total positivity of the almost-Riordan arrays. The production matrix J of an almost-Riordan array ( d \| g , f ) is presented so that the total positivity of J implies that of both the almost-Riordan array ( d \| g , f ) and the Riordan array ( g , f ). We also present a counterexample to illustrate that this sufficient condition is not necessary. If the production matrix J is tridiagonal, then the expressions of its principal minors are given. By using these expressions, we find a sufficient and necessary condition of the total positivity of almost-Riordan arrays with tridiagonal production matrices. Numerous examples are given to demonstrate our results.
AbstractList	In this paper we study the total positivity of almost-Riordan arrays ( d ( t ) \| g ( t ) , f ( t ) ) and establish its necessary conditions and sufficient conditions, particularly, for some well-used formal power series d ( t ). We present a semidirect product of an almost-array and use it to transfer a total positivity problem for an almost-Riordan array to the total positivity problem for a quasi-Riordan array. We find the sequence characterization of total positivity of the almost-Riordan arrays. The production matrix J of an almost-Riordan array ( d \| g , f ) is presented so that the total positivity of J implies that of both the almost-Riordan array ( d \| g , f ) and the Riordan array ( g , f ). We also present a counterexample to illustrate that this sufficient condition is not necessary. If the production matrix J is tridiagonal, then the expressions of its principal minors are given. By using these expressions, we find a sufficient and necessary condition of the total positivity of almost-Riordan arrays with tridiagonal production matrices. Numerous examples are given to demonstrate our results. In this paper we study the total positivity of almost-Riordan arrays (d(t)\|g(t),f(t)) and establish its necessary conditions and sufficient conditions, particularly, for some well-used formal power series d(t). We present a semidirect product of an almost-array and use it to transfer a total positivity problem for an almost-Riordan array to the total positivity problem for a quasi-Riordan array. We find the sequence characterization of total positivity of the almost-Riordan arrays. The production matrix J of an almost-Riordan array (d\|g,f) is presented so that the total positivity of J implies that of both the almost-Riordan array (d\|g,f) and the Riordan array (g, f). We also present a counterexample to illustrate that this sufficient condition is not necessary. If the production matrix J is tridiagonal, then the expressions of its principal minors are given. By using these expressions, we find a sufficient and necessary condition of the total positivity of almost-Riordan arrays with tridiagonal production matrices. Numerous examples are given to demonstrate our results.
ArticleNumber	115
Author	He, Tian-Xiao Słowik, Roksana
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Cites_doi	10.1016/j.laa.2019.01.015 10.1016/0097-3165(95)90000-4 10.1016/j.laa.2023.01.012 10.1016/j.laa.2020.02.021 10.1016/j.ejc.2014.11.009 10.1073/pnas.37.5.303 10.1016/j.laa.2021.11.005 10.1007/BF02786970
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Subtitle	Total Positivity of Almost-Riordan Arrays
Title	Total Positivity of Almost-Riordan Arrays
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