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

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.