Total Positivity of Almost-Riordan Arrays
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911907970023424 |
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| author | He, Tian-Xiao Słowik, Roksana |
| author_facet | He, Tian-Xiao Słowik, Roksana |
| contents | 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 $J$ is totally positive implies the total positivity 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 expressions, we find a sufficient and necessary condition of the total positivity of almost-Riordan arrays with tridiagonal production matrices. A numerous examples are given to demonstrate our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03774 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Total Positivity of Almost-Riordan Arrays He, Tian-Xiao Słowik, Roksana Combinatorics 05A15, 05A05, 15B36, 15A06, 05A19, 11B83 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 $J$ is totally positive implies the total positivity 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 expressions, we find a sufficient and necessary condition of the total positivity of almost-Riordan arrays with tridiagonal production matrices. A numerous examples are given to demonstrate our results. |
| title | Total Positivity of Almost-Riordan Arrays |
| topic | Combinatorics 05A15, 05A05, 15B36, 15A06, 05A19, 11B83 |
| url | https://arxiv.org/abs/2406.03774 |