Infinite log-concavity: developments and conjectures
Given a sequence $(a_k)=a_0,a_1,a_2,\ldots$ of real numbers, define a new sequence $\mathcal{L}(a_k)=(b_k)$ where $b_k=a_k^2-a_{k-1}a_{k+1}$. So $(a_k)$ is log-concave if and only if $(b_k)$ is a nonnegative sequence. Call $(a_k)$ $\textit{infinitely log-concave}$ if $\mathcal{L}^i(a_k)$ is nonnegat...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
2009-01-01
|
| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/2678/pdf |
| _version_ | 1797270361211928576 |
|---|---|
| author | Peter R. W. McNamara Bruce E. Sagan |
| author_facet | Peter R. W. McNamara Bruce E. Sagan |
| author_sort | Peter R. W. McNamara |
| collection | DOAJ |
| description | Given a sequence $(a_k)=a_0,a_1,a_2,\ldots$ of real numbers, define a new sequence $\mathcal{L}(a_k)=(b_k)$ where $b_k=a_k^2-a_{k-1}a_{k+1}$. So $(a_k)$ is log-concave if and only if $(b_k)$ is a nonnegative sequence. Call $(a_k)$ $\textit{infinitely log-concave}$ if $\mathcal{L}^i(a_k)$ is nonnegative for all $i \geq 1$. Boros and Moll conjectured that the rows of Pascal's triangle are infinitely log-concave. Using a computer and a stronger version of log-concavity, we prove their conjecture for the $n$th row for all $n \leq 1450$. We can also use our methods to give a simple proof of a recent result of Uminsky and Yeats about regions of infinite log-concavity. We investigate related questions about the columns of Pascal's triangle, $q$-analogues, symmetric functions, real-rooted polynomials, and Toeplitz matrices. In addition, we offer several conjectures. |
| first_indexed | 2024-04-25T02:03:03Z |
| format | Article |
| id | doaj.art-8653abbcbbb34409a86064df146f7b59 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:03:03Z |
| publishDate | 2009-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-8653abbcbbb34409a86064df146f7b592024-03-07T14:45:40ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502009-01-01DMTCS Proceedings vol. AK,...Proceedings10.46298/dmtcs.26782678Infinite log-concavity: developments and conjecturesPeter R. W. McNamaraBruce E. Sagan0Department of Mathematics [Lansing]Given a sequence $(a_k)=a_0,a_1,a_2,\ldots$ of real numbers, define a new sequence $\mathcal{L}(a_k)=(b_k)$ where $b_k=a_k^2-a_{k-1}a_{k+1}$. So $(a_k)$ is log-concave if and only if $(b_k)$ is a nonnegative sequence. Call $(a_k)$ $\textit{infinitely log-concave}$ if $\mathcal{L}^i(a_k)$ is nonnegative for all $i \geq 1$. Boros and Moll conjectured that the rows of Pascal's triangle are infinitely log-concave. Using a computer and a stronger version of log-concavity, we prove their conjecture for the $n$th row for all $n \leq 1450$. We can also use our methods to give a simple proof of a recent result of Uminsky and Yeats about regions of infinite log-concavity. We investigate related questions about the columns of Pascal's triangle, $q$-analogues, symmetric functions, real-rooted polynomials, and Toeplitz matrices. In addition, we offer several conjectures.https://dmtcs.episciences.org/2678/pdfreal rootsbinomial coefficientscomputer proofgaussian polynomialinfinite log-concavitysymmetric functions[math.math-co] mathematics [math]/combinatorics [math.co][info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Peter R. W. McNamara Bruce E. Sagan Infinite log-concavity: developments and conjectures Discrete Mathematics & Theoretical Computer Science real roots binomial coefficients computer proof gaussian polynomial infinite log-concavity symmetric functions [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Infinite log-concavity: developments and conjectures |
| title_full | Infinite log-concavity: developments and conjectures |
| title_fullStr | Infinite log-concavity: developments and conjectures |
| title_full_unstemmed | Infinite log-concavity: developments and conjectures |
| title_short | Infinite log-concavity: developments and conjectures |
| title_sort | infinite log concavity developments and conjectures |
| topic | real roots binomial coefficients computer proof gaussian polynomial infinite log-concavity symmetric functions [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2678/pdf |
| work_keys_str_mv | AT peterrwmcnamara infinitelogconcavitydevelopmentsandconjectures AT bruceesagan infinitelogconcavitydevelopmentsandconjectures |