The Real-rootedness of Generalized Narayana Polynomials

The Real-rootedness of Generalized Narayana Polynomials
复制标题

DOI：
10.1216/rmj-2018-48-1-107
复制
发表时间：
2016-02
期刊：
arXiv: Combinatorics
影响因子：
--
通讯作者：
Herman Z. Q. Chen;A. Yang;Philip B. Zhang
Herman Z. Q. Chen;A. Yang;Philip B. Zhang
中科院分区：
其他
文献类型：
--
作者：
Herman Z. Q. Chen;A. Yang;Philip B. Zhang

文献摘要

被引文献

相似文献

本文证明了两类广义Narayana多项式的实根性：一类是有限Weyl群的广义关联面体的$h$ -多项式，另一类是Boros-Moll多项式的无限对数凹性的研究。对于前者，Brändén已经证明了这些$h$ -多项式只有实零。建立了两类Narayana多项式的递推关系，并由此导出了多项式的实根性。为了证明多项式的实根性，我们利用Liu和Wang给出的一个充分条件来确定两个多项式是否有交错的零。利用Mathematica软件包\textit{HolonomicFunctions}对递归关系进行了验证。
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}.