Involve: A Journal of Mathematics

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

Zeros of polynomials with four-term recurrence

Khang Tran and Andres Zumba

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Abstract

Given real numbers b,c, 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 H1(z)=H2(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=0Z(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
Received: 14 March 2017
Revised: 7 June 2017
Accepted: 19 June 2017
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Subjects
Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05] 11C08: Polynomials [See also 13F20]

Keywords
generating functions hyperbolic polynomials recursive sequence

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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