Yang, Arthur L.B. and Zhang, Philip B. - The Real-rootedness of Eulerian Polynomials via the Hermite–Biehler Theorem

dmtcs:2510 - Discrete Mathematics & Theoretical Computer Science, January 1, 2015, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
The Real-rootedness of Eulerian Polynomials via the Hermite–Biehler Theorem

Authors: Yang, Arthur L.B. and Zhang, Philip B.

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.

Source : oai:HAL:hal-01337801v1
Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Section: Proceedings
Published on: January 1, 2015
Submitted on: November 21, 2016
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]