Infinite log-concavity: developments and conjectures

TL;DR

This paper proves that the rows of Pascal's triangle are infinitely log-concave for all n ≤ 1450, using computational methods and explores related properties and conjectures in combinatorics.

Contribution

It confirms Boros and Moll's conjecture for a large range of Pascal's triangle rows and introduces new methods for analyzing infinite log-concavity.

Findings

Proved infinite log-concavity for Pascal's triangle rows up to n=1450

Provided a simple proof for a recent result on regions of infinite log-concavity

Explored related questions in q-analogues, symmetric functions, and Toeplitz matrices

Abstract

Given a sequence (a_k) = a_0, a_1, a_2,... of real numbers, define a new sequence 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) "infinitely log-concave" if L^i(a_k) is nonnegative for all i >= 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 nth row for all n <= 1450. We 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.