## Involve: A Journal of Mathematics

• Involve
• Volume 11, Number 3 (2018), 501-518.

### Zeros of polynomials with four-term recurrence

#### Abstract

Given real numbers $b,c∈ℝ$, we form the sequence of polynomials ${Hm(z)}m=0∞$ satisfying the four-term recurrence

$H m ( z ) + c H m − 1 ( z ) + b H m − 2 ( z ) + z H m − 3 ( z ) = 0 , m ≥ 1 ,$

with the initial conditions $H0(z)=1$ and $H−1(z)=H−2(z)=0$. We find necessary and sufficient conditions on $b$ and $c$ under which the zeros of $Hm(z)$ are real for all $m$, and provide an explicit real interval on which $⋃m=0∞Z(Hm)$ is dense, where $Z(Hm)$ is the set of zeros of $Hm(z)$.

#### Article information

Source
Involve, Volume 11, Number 3 (2018), 501-518.

Dates
Revised: 7 June 2017
Accepted: 19 June 2017
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513775080

Digital Object Identifier
doi:10.2140/involve.2018.11.501

Mathematical Reviews number (MathSciNet)
MR3733971

Zentralblatt MATH identifier
06817034

#### Citation

Tran, Khang; Zumba, Andres. Zeros of polynomials with four-term recurrence. Involve 11 (2018), no. 3, 501--518. doi:10.2140/involve.2018.11.501. https://projecteuclid.org/euclid.involve/1513775080

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