On the positive zeros of generalized Narayana polynomials related to the Boros-Moll polynomials

TL;DR

This paper investigates the positive zeros of generalized Narayana polynomials related to Boros-Moll polynomials, establishing bounds and monotonicity properties using a new recurrence relation.

Contribution

It introduces a new recurrence relation for these polynomials and proves bounds and monotonicity of their positive zeros.

Findings

Established upper and lower bounds for positive zeros.

Proved monotonicity of positive zeros with respect to parameters.

Connected properties of zeros to the recurrence relation.

Abstract

The generalized Narayana polynomials $N_{n,m}(x)$ arose from the study of infinite log-concavity of the Boros-Moll polynomials. The real-rootedness of $N_{n,m}(x)$ had been proved by Chen, Yang and Zhang. They also showed that when $n\geq m+2$, each of the generalized Narayana polynomials has one and only one positive zero and $m$ negative zeros, where the negative zeros of $N_{n,m}(x)$ and $N_{n+1,m+1}(x)$ have interlacing relations. In this paper, we study the properties of the positive zeros of $N_{n,m}(x)$ for $n\geq m+2$. We first obtain a new recurrence relation for the generalized Narayana polynomials. Based on this recurrence relation, we prove upper and lower bounds for the positive zeros of $N_{n,m}(x)$. Moreover, the monotonicity of the positive zeros of $N_{n,m}(x)$ are also proved by using the new recurrence relation.