PreprintCC BY 4.0arXiv
k-Lucas Annulus for Polynomial Zeros
Abstract
We prove a binomial identity relating the k-Lucas and k-Fibonacci sequences: for real k>0 and integers m≥1, n≥1, a binomial-weighted sum of k-Lucas numbers reduces to a multiple of Lk,mn when n is even, but to a multiple of Fk,mn when n is odd. This parity dependence traces back to the k-Lucas Binet formula lacking the normalising factor 1/(α−β) present for k-Fibonacci numbers. The identity supplies, for each parity of n and each m, an explicit family of positive weights summing to one; combined with the general annulus principle of Dalal and Govil, this yields a k-Lucas annulus containing all the zeros of a complex polynomial of degree n, complementing the k-Fibonacci-based bounds of Díaz-Barrero, Bidkham-Shashahani, and Kaur.
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BibTeX
@misc{batte2026klucas,
title = {{$k$-Lucas Annulus for Polynomial Zeros}},
author = {Herbert Batte},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.26452},
eprint = {2608.26452},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26452},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - k-Lucas Annulus for Polynomial Zeros
AU - Herbert Batte
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2608.26452
UR - https://arxiv.org/abs/2608.26452
AB - We prove a binomial identity relating the k-Lucas and k-Fibonacci sequences: for real k>0 and integers m≥1, n≥1, a binomial-weighted sum of k-Lucas numbers reduces to a multiple of L_k,mn when n is even, but to a multiple of F_k,mn when n is odd. This parity dependence traces back to the k-Lucas Binet formula lacking the normalising factor 1/(α−β) present for k-Fibonacci numbers. The identity supplies, for each parity of n and each m, an explicit family of positive weights summing to one; combined with the general annulus principle of Dalal and Govil, this yields a k-Lucas annulus containing all the zeros of a complex polynomial of degree n, complementing the k-Fibonacci-based bounds of Díaz-Barrero, Bidkham-Shashahani, and Kaur.
ER -