Involve: A Journal of Mathematics

  • Involve
  • Volume 8, Number 2 (2015), 211-220.

Convergence of the maximum zeros of a class of Fibonacci-type polynomials

Rebecca Grider and Kristi Karber

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Abstract

Let a be a positive integer and let k be an arbitrary, fixed positive integer. We define a generalized Fibonacci-type polynomial sequence by Gk,0(x) = a, Gk,1(x) = x a, and Gk,n(x) = xkGk,n1(x) + Gk,n2(x) for n 2. Let gk,n represent the maximum real zero of Gk,n. We prove that the sequence {gk,2n} is decreasing and converges to a real number βk. Moreover, we prove that the sequence {gk,2n+1} is increasing and converges to βk as well. We conclude by proving that {βk} is decreasing and converges to a.

Article information

Source
Involve, Volume 8, Number 2 (2015), 211-220.

Dates
Received: 7 October 2012
Revised: 16 June 2013
Accepted: 19 October 2013
First available in Project Euclid: 22 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1511370856

Digital Object Identifier
doi:10.2140/involve.2015.8.211

Mathematical Reviews number (MathSciNet)
MR3320854

Zentralblatt MATH identifier
1310.11022

Subjects
Primary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
Secondary: 11B37: Recurrences {For applications to special functions, see 33-XX} 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}

Keywords
Fibonacci polynomial convergence zeros roots

Citation

Grider, Rebecca; Karber, Kristi. Convergence of the maximum zeros of a class of Fibonacci-type polynomials. Involve 8 (2015), no. 2, 211--220. doi:10.2140/involve.2015.8.211. https://projecteuclid.org/euclid.involve/1511370856


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