Zeros of polynomials with four-term recurrence and linear coefficients

TL;DR

This paper analyzes the zero distribution of polynomials generated by a specific cubic reciprocal, establishing conditions for real zeros and identifying dense intervals containing these zeros.

Contribution

It provides necessary and sufficient conditions for the reality of zeros of these polynomials and describes the structure of the intervals containing them.

Findings

Identifies conditions for real zeros of the polynomial sequence.

Determines explicit intervals containing the zeros.

Shows the union of these intervals is dense within a certain range.

Abstract

This paper investigates the zero distribution of a sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ generated by the reciprocal of $1+ct+B(z)t^{2}+A(z)t^{3}$ where $c\in\mathbb{R}$ and $A(z)$, $B(z)$ are real linear polynomials. We study necessary and sufficient conditions for the reality of the zeros of $P_{m}(z)$. Under these conditions, we find an explicit interval containing these zeros, whose union forms a dense subset of this interval.