# Infinite log-concavity: developments and conjectures

Abstract : 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.
Keywords :
Document type :
Conference papers
Domain :
Complete list of metadata

Cited literature [25 references]

https://hal.inria.fr/hal-01185370
Contributor : Coordination Episciences Iam Connect in order to contact the contributor
Submitted on : Thursday, August 20, 2015 - 11:06:11 AM
Last modification on : Wednesday, August 7, 2019 - 2:32:05 PM
Long-term archiving on: : Wednesday, April 26, 2017 - 10:18:35 AM

### File

dmAK0153.pdf
Publisher files allowed on an open archive

### Citation

Peter R. W. Mcnamara, Bruce E. Sagan. Infinite log-concavity: developments and conjectures. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.635-646, ⟨10.46298/dmtcs.2678⟩. ⟨hal-01185370⟩

Record views