Analytic properties of sextet polynomials of hexagonal systems

TL;DR

This paper studies the analytic properties of sextet polynomials in hexagonal systems, revealing real zeros, symmetry, unimodality, and conjecturing log-concavity, with implications for understanding their mathematical structure.

Contribution

It provides new insights into the zeros and coefficient properties of sextet polynomials for hexagonal systems, including pyrene chains, and proposes conjectures for general cases.

Findings

Zeros of sextet polynomials for pyrene chains are real and dense in a specific interval.

Coefficients of these polynomials are symmetric, unimodal, log-concave, and asymptotically normal.

Real zeros of sextet polynomials for all hexagonal systems are dense in , and log-concavity is conjectured.

Abstract

In this paper we investigate analytic properties of sextet polynomials of hexagonal systems. For the pyrene chains, we show that zeros of the sextet polynomials $P_n(x)$ are real, located in the open interval $(-3-2\sqrt{2},-3+2\sqrt{2})$ and dense in the corresponding closed interval. We also show that coefficients of $P_n(x)$ are symmetric, unimodal, log-concave, and asymptotically normal. For general hexagonal systems, we show that real zeros of all sextet polynomials are dense in the interval $(-\infty,0]$, and conjecture that every sextet polynomial has log-concave coefficients.