The Ratio Monotonicity of the Boros-Moll Polynomials

TL;DR

This paper proves that Boros-Moll polynomials have the ratio monotone property, which implies their log-concavity and spiral property, and introduces a stronger conjecture related to their infinite log-concavity.

Contribution

It establishes the ratio monotone property for Boros-Moll polynomials, advancing understanding of their structural properties and proposing a new conjecture on their infinite log-concavity.

Findings

Boros-Moll polynomials are ratio monotone.

Ratio monotonicity implies log-concavity and spiral property.

Proposes a stronger conjecture on infinite log-concavity.

Abstract

In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are log-concave. In this paper, we show that the Boros-Moll polynomials possess the ratio monotone property which implies the log-concavity and the spiral property. We conclude with a conjecture which is stronger than Moll's conjecture on the $\infty$-log-concavity.