• Corpus ID: 10041963

The Real-rootedness of Generalized Narayana Polynomials

@article{Chen2016TheRO,
  title={The Real-rootedness of Generalized Narayana Polynomials},
  author={Herman Z. Q. Chen and Arthur L. B. Yang and Philip B. Zhang},
  journal={arXiv: Combinatorics},
  year={2016}
}

• Herman Z. Q. Chen, Arthur L. B. Yang, Philip B. Zhang
• Published 1 February 2016
• Mathematics
• arXiv: Combinatorics

In this paper, we prove the real-rootedness of two classes of generalized Narayana polynomials: one arising as the $h$-polynomials of the generalized associahedron associated to the finite Weyl groups, the other arising in the study of the infinite log-concavity of the Boros-Moll polynomials. For the former, Br\"{a}nd\'{e}n has already proved that these $h$-polynomials have only real zeros. We establish certain recurrence relations for the two classes of Narayana polynomials, from which we derive the real-rootedness. To prove the real-rootedness, we use a sufficient condition, due to Liu and Wang, to determine whether two polynomials have interlaced zeros. The recurrence relations are verified with the help of the Mathematica package \textit{HolonomicFunctions}.