Polynomials with only real zeros and the Eulerian polynomials of type D

TL;DR

This paper proves Brenti's conjecture that Eulerian polynomials of type D have only real zeros, by exploring an equivalent form of a key identity involving these polynomials.

Contribution

It establishes the real-rootedness of Eulerian polynomials of type D, confirming a conjecture and deepening understanding of their properties.

Findings

Proves Brenti's real-rootedness conjecture for type D Eulerian polynomials

Provides an equivalent form of a known identity involving these polynomials

Enhances understanding of the roots and structure of type D Eulerian polynomials

Abstract

A remarkable identity involving the Eulerian polynomials of type D was obtained by Stembridge (Adv. Math. 106 (1994), p. 280, Lemma 9.1). In this paper we explore an equivalent form of this identity. We prove Brenti's real-rootedness conjecture for the Eulerian polynomials of type $D$.