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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 49 Issue: 5

A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences

Fikri Köken(Necmettin Erbakan University)0000-0002-8304-9525
Sayfa
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Yayın Tarihi
2020-10-06
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

Many combinatorial identities are derived as applications.

📌 3 Maddede Bu Makale

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Amaç & Hipotez: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.
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Temel Bulgular & Veri: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.
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Sonuç & Katkı:Many combinatorial identities are derived as applications.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler: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 as applications.

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Bu Makaleye Atıf Yap

Köken, F. (2020). A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 5. https://doi.org/10.15672/hujms.481026

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