The Real-rootedness of Generalized Narayana Polynomials

TL;DR

This paper proves the real-rootedness of two classes of generalized Narayana polynomials, one related to Weyl groups and the other to Boros-Moll polynomials, using recurrence relations and interlacing zero criteria.

Contribution

It establishes new recurrence relations for these polynomials and applies interlacing criteria to prove their real-rootedness, extending previous results.

Findings

Proved real-rootedness of generalized Narayana polynomials.

Derived recurrence relations for these polynomials.

Validated interlacing conditions using computational tools.

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, 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}.