The Real-rootedness of Eulerian Polynomials via the Hermite--Biehler Theorem

TL;DR

This paper proves the real-rootedness of certain Eulerian polynomials using the Hermite--Biehler theorem, confirming conjectures and extending understanding of polynomial stability properties.

Contribution

It introduces a new proof technique for polynomial real-rootedness based on the Hermite--Biehler theorem, expanding the theoretical framework for Eulerian polynomials.

Findings

Proved real-rootedness of type D Eulerian polynomials.

Confirmed Hyatt's conjectures on interlacing of half Eulerian polynomials.

Extended stability analysis using Borcea and Brändén's operator characterization.

Abstract

Based on the Hermite--Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $\mathbf{s}$-Eulerian polynomials. We also confirm Hyatt's conjectures on the interlacing property of half Eulerian polynomials. Borcea and Br\"and\'en's work on the characterization of linear operators preserving Hurwitz stability is critical to this approach.