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

#### Fikri KÖKEN [1]

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.
generalized Fibonacci and Lucas sequences, generalized Fibonacci and Lucas matrices, Pascal matrices
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Primary Language en Mathematics Mathematics Orcid: 0000-0002-8304-9525Author: Fikri KÖKEN (Primary Author)Institution: Necmettin Erbakan UniversityCountry: Turkey Publication Date : October 6, 2020
 Bibtex @research article { hujms481026, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1735 - 1743}, doi = {10.15672/hujms.481026}, title = {A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences}, key = {cite}, author = {Köken, Fikri} } APA Köken, F . (2020). A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences . Hacettepe Journal of Mathematics and Statistics , 49 (5) , 1735-1743 . DOI: 10.15672/hujms.481026 MLA Köken, F . "A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1735-1743 Chicago Köken, F . "A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1735-1743 RIS TY - JOUR T1 - A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences AU - Fikri Köken Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.481026 DO - 10.15672/hujms.481026 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1735 EP - 1743 VL - 49 IS - 5 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.481026 UR - https://doi.org/10.15672/hujms.481026 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences %A Fikri Köken %T A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 5 %R doi: 10.15672/hujms.481026 %U 10.15672/hujms.481026 ISNAD Köken, Fikri . "A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences". Hacettepe Journal of Mathematics and Statistics 49 / 5 (October 2020): 1735-1743 . https://doi.org/10.15672/hujms.481026 AMA Köken F . A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences. Hacettepe Journal of Mathematics and Statistics. 2020; 49(5): 1735-1743. Vancouver Köken F . A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences. Hacettepe Journal of Mathematics and Statistics. 2020; 49(5): 1735-1743.

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