Zeros of polynomials with four-term recurrence

Abstract

For any real numbers $b,c\in \mathbb{R}$, we form the sequence of polynomials ${\left\{H_m(z)\right\}}_{m=0}^{\infty }$ satisfying the four-term recurrence \[ H_{m}(z)+cH_{m-1}(z)+bH_{m-2}(z)+zH_{m-3}(z)=0,\qquad m\ge3, \] with the initial conditions $H_0(z)=1$, $H_1(z)=$$-c$, and $H_2(z)=-b+c^2$. We find necessary and sufficient conditions on $b$ and $c$ under which the zeros of $H_m(z)$ are real for all $m$, and provide an explicit real interval on which $\bigcup _{m=0}^{\infty }\mathcal{Z}(H_m)$ is dense where $\mathcal{Z}(H_m)$ is the set of zeros of $H_m(z)$.

References in corpus (5)

• Connections between discriminants and the root distribution of polynomials with rational generating function
• On rational approximation of algebraic functions
• The root distribution of polynomials with a three-term recurrence
• Polynomials with rational generating functions and real zeros
• Non-Real Zero Decreasing Operators Related to Orthogonal Polynomials