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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Bibliographic Details
Published in:Graphs and combinatorics Vol. 41; no. 6; p. 115
Main Authors: He, Tian-Xiao, Słowik, Roksana
Format: Journal Article
Language:English
Published: Tokyo Springer Japan 01.12.2025
Springer Nature B.V
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-025-02979-6