Insights Into the Lucas Q-Matrix and Its Properties

Authors

  • Maria Eduarda dos Santos Chaves Quirino Universidade Estadual Paulista “Julio de Mesquita Filho”, UNESP, Rio Claro, Sao Paulo, Brasil https://orcid.org/0009-0008-7673-041X
  • Lucas Antonio Caritá Instituto Federal de Educacao, Ciencia e Tecnologia de Sao Paulo, IFSP, Grupo de Pesquisa em Matematica Cientıfica e Computacional - GPMCC, Sao Jose dos Campos, Sao Paulo, Brasil https://orcid.org/0000-0002-9518-3414

DOI:

https://doi.org/10.26713/jims.v17i1.3023

Abstract

This work proves that the equality
[3112]n={5n2[Fn+1FnFnFn1],if n is even,5n12[Ln+1LnLnLn1],if n is odd,
holds for all integer n, where [3112] is the Lucas Q-matrix introduced by Köken and Bozkurt [2]. While these authors established the case for natural n using induction, extending the result to integer exponents is considerably more challenging. One cannot simply assume that the formula holds for an arbitrary integer n, since, although Fibonacci and Lucas numbers are defined for negative indices, the proof by mathematical induction does not automatically extend to this setting. To achieve this, we developed several key prerequisites, including new relationships between the Fibonacci and Lucas Q-matrices, such as [Ln+1LnLnLn1]=[3112][FnFn1Fn1Fn2]=QLQFn1, nZ. Additionally, we demonstrate several previously known properties, but in a different way, using our properties. Because this text is also a survey article, we adopted a didactic, step-by-step approach, minimizing omissions whenever possible.

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Published

2025-04-29
CITATION

How to Cite

Quirino, M. E. dos S. C., & Caritá, L. A. (2025). Insights Into the Lucas Q-Matrix and Its Properties. Journal of Informatics and Mathematical Sciences, 17(1), 21–41. https://doi.org/10.26713/jims.v17i1.3023

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Section

Research Article