Many combinatorial identities are derived asapplications.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:In this study, a matrix $R_{v}$ is defined, and two closed form expressions of the matrix $R_{v}^{n}$, for an integer $n\geq 1$, are evaluated by the matrix functions in matrix theory.
📊
Key Findings & Data:By modifying results of $2\times 2$ matrix representations given in the references of our study, we give various $3\times 3$ matrix representations of the generalized Fibonacci and Lucas sequences.
💡
Conclusion & Impact:Many combinatorial identities are derived asapplications.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:generalized Fibonacci and Lucas sequences · generalized Fibonacci and Lucas matrices · Pascal matrices
Abstract
In this study, a matrix $R_{v}$ is defined, and two closed form expressions of the matrix $R_{v}^{n}$, for an integer $n\geq 1$, are evaluated by the matrix functions in matrix theory. These expressions satisfy a connection between the generalized Fibonacci and Lucas numbers with the Pascal matrices. Thus, two representations of the matrix $R_{v}^{n}$ and various forms of matrix $(R_{v}+q\triangle I)^{n}$ are studied in terms of the generalized Fibonacci and Lucas numbers and binomial coefficients. By modifying results of $2\times 2$ matrix representations given in the references of our study, we give various $3\times 3$ matrix representations of the generalized Fibonacci and Lucas sequences. Many combinatorial identities are derived asapplications.
KÖKEN, F. (2022). A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 5, 1735–1743. https://ajindex.com/articles/529623