Distribution of the roots of Eulerian polynomials

TL;DR

This paper proves that the roots of Eulerian polynomials asymptotically follow a log-Cauchy distribution by analyzing their moments, which become constant and relate to Cauchy numbers of the second kind.

Contribution

It provides a new proof of the root distribution convergence and links the moments of roots to Cauchy numbers, revealing their asymptotic behavior.

Findings

Roots' empirical measures converge to a log-Cauchy distribution.

Moments of roots become constant and are expressed via Cauchy numbers.

Asymptotic root distribution is characterized mathematically.

Abstract

We give a new proof that the empirical measures of the roots of Eulerian polynomials converge to a certain log-Cauchy distribution. To do so, we show that each moment of the roots of a related family of polynomials not only converge, but in fact become ultimately constant. These asymptotic moments are expressed in terms of Cauchy numbers of the second kind.