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

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 $s$-Eulerian polynomials. We also confirm Hyatt’s conjectures on the inter-lacing property of half Eulerian polynomials. Borcea and Brändén’s work on the characterization of linear operators preserving Hurwitz stability is critical to this approach.

Keywords

• eulerian polynomials
• hermite–biehler theorem
• borcea and brändén’s stability criterion
• weak hurwitz stability
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]