The Real-rootedness of Generalized Narayana Polynomials

Abstract

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, Brändén 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}.

arXiv comment14 pages

References in corpus (2)

• Generalized cluster complexes and Coxeter combinatorics
• Questions about determinants and polynomials

Cited by in corpus (1)

• Total positivity of Narayana matrices