## Abstract

Let {a_{n}}_{n≥0} denote the linear recursive sequence of order k (k ≥ 2) defined by the initial values a_{0} = a_{1} = · · · = a_{k-2} = 0 and a_{k-1} = 1 and the recursion a_{n} = a_{n-1} + a_{n-2} + · · · + a_{n-k} if n ≥ k. The an are often called k-Fibonacci numbers and reduce to the usual Fibonacci numbers when k = 2. Let P_{n,k}(x) = a_{k-1}x^{n} + a_{k}x_{n-1} + · · · + a_{n+k}-2^{x} + a_{n+k-1}, which we will refer to as a k-Fibonacci coefficient polynomial. In this paper, we show for all k that the polynomial P_{n,k}(x) has no real zeros if n is even and exactly one real zero if n is odd. This generalizes the known result for the k = 2 and k = 3 cases corresponding to Fibonacci and Tribonacci coefficient polynomials, respectively. It also improves upon a previous upper bound of approximately k for the number of real zeros of P_{n,k}(x). Finally, we show for all k that the sequence of real zeros of the polynomials P_{n,k}(x) when n is odd converges to the opposite of the positive zero of the characteristic polynomial associated with the sequence an. This generalizes a previous result for the case k = 2.

Original language | English |
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Pages (from-to) | 57-76 |

Number of pages | 20 |

Journal | Annales Mathematicae et Informaticae |

Volume | 40 |

State | Published - 2012 |

## Keywords

- K-fibonacci sequence
- Linear recurrences
- Zeros of polynomials

## ASJC Scopus subject areas

- General Computer Science
- General Mathematics