Vol. 11, No. 3, 2018

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Zeros of polynomials with four-term recurrence

Khang Tran and Andres Zumba

Vol. 11 (2018), No. 3, 501–518
Abstract

Given real numbers $b,c\in ℝ$, we form the sequence of polynomials ${\left\{{H}_{m}\left(z\right)\right\}}_{m=0}^{\infty }$ satisfying the four-term recurrence

${H}_{m}\left(z\right)+c{H}_{m-1}\left(z\right)+b{H}_{m-2}\left(z\right)+z{H}_{m-3}\left(z\right)=0,\phantom{\rule{1em}{0ex}}m\ge 1,$

with the initial conditions ${H}_{0}\left(z\right)=1$ and ${H}_{-1}\left(z\right)={H}_{-2}\left(z\right)=0$. We find necessary and sufficient conditions on $b$ and $c$ under which the zeros of ${H}_{m}\left(z\right)$ are real for all $m$, and provide an explicit real interval on which ${\bigcup }_{m=0}^{\infty }\mathsc{Z}\left({H}_{m}\right)$ is dense, where $\mathsc{Z}\left({H}_{m}\right)$ is the set of zeros of ${H}_{m}\left(z\right)$.

Keywords
generating functions, hyperbolic polynomials, recursive sequence
Mathematical Subject Classification 2010
Primary: 30C15, 26C10, 11C08