Infinite Log-concavity: Developments and Conjectures

Peter McNamara; Bruce Sagan

Infinite Log-concavity: Developments and Conjectures

Peter McNamara, Bruce Sagan

Bucknell University

Michigan State University

Research output: Contribution to journal › Article › peer-review

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 . 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 &ges;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 &les;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.

Original language	American English
Journal	Default journal
Volume	44
State	Published - Jan 1 2010

Keywords

binomial coefficients
computer proof
Gaussian polynomial
infinite log-concavity
real roots
symmetric functions
toeplitz matrices

Disciplines

Mathematics

Access to Document

http://www.facstaff.bucknell.edu/pm040/Papers/ilc.pdf

Cite this

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title = "Infinite Log-concavity: Developments and Conjectures",

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 . 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 \&ges;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 \&les;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.",

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T1 - Infinite Log-concavity: Developments and Conjectures

AU - McNamara, Peter

AU - Sagan, Bruce

PY - 2010/1/1

Y1 - 2010/1/1

N2 - Given a sequence ( a k )= a 0 , a 1 , a 2 ,… of real numbers, define a new sequence L ( a k )=( b k ) where . 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 &ges;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 &les;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.

AB - Given a sequence ( a k )= a 0 , a 1 , a 2 ,… of real numbers, define a new sequence L ( a k )=( b k ) where . 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 &ges;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 &les;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.

KW - binomial coefficients

KW - computer proof

KW - Gaussian polynomial

KW - infinite log-concavity

KW - real roots

KW - symmetric functions

KW - toeplitz matrices

UR - https://digitalcommons.bucknell.edu/fac_journ/116

UR - http://www.facstaff.bucknell.edu/pm040/Papers/ilc.pdf

M3 - Article

VL - 44

JO - Default journal

JF - Default journal

ER -

Infinite Log-concavity: Developments and Conjectures

Abstract

Keywords

Disciplines

Access to Document

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