On hyper-dual vectors and angles with Pell, Pell-Lucas numbers

Faik Babadağ; Ali Atasoy; Faik Babadağ; Ali Atasoy

doi:10.3934/math.20241480

AIMS Mathematics

2024, Volume 9, Issue 11: 30655-30666. doi: 10.3934/math.20241480

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Research article

On hyper-dual vectors and angles with Pell, Pell-Lucas numbers

Faik Babadağ ^{1
,
,},
Ali Atasoy ²

1.
Department of Mathematics, Kırıkkale University, 71450, Yahşihan, Kırıkkale, Turkey
2.
Keskin Vocational School, Kırıkkale University, 71800, Keskin, Kırıkkale, Turkey

Received: 25 July 2024 Revised: 12 September 2024 Accepted: 12 October 2024 Published: 28 October 2024
MSC : 11B37, 53A45

In this paper, we introduce two types of hyper-dual numbers with components including Pell and Pell-Lucas numbers. This novel approach facilitates our understanding of hyper-dual numbers and properties of Pell and Pell-Lucas numbers. We also investigate fundamental properties and identities associated with Pell and Pell-Lucas numbers, such as the Binet-like formulas, Vajda-like, Catalan-like, Cassini-like, and d'Ocagne-like identities. Furthermore, we also define hyper-dual vectors by using Pell and Pell-Lucas vectors and discuse hyper-dual angles. This extensionis not only dependent on our understanding of dual numbers, it also highlights the interconnectedness between integer sequences and geometric concepts.
- hyper-dual Pell number,
- hyper-dual Pell-Lucas number,
- hyper-dual Pell vector,
- hyper-dual angle
Citation: Faik Babadağ, Ali Atasoy. On hyper-dual vectors and angles with Pell, Pell-Lucas numbers[J]. AIMS Mathematics, 2024, 9(11): 30655-30666. doi: 10.3934/math.20241480

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Abstract

In this paper, we introduce two types of hyper-dual numbers with components including Pell and Pell-Lucas numbers. This novel approach facilitates our understanding of hyper-dual numbers and properties of Pell and Pell-Lucas numbers. We also investigate fundamental properties and identities associated with Pell and Pell-Lucas numbers, such as the Binet-like formulas, Vajda-like, Catalan-like, Cassini-like, and d'Ocagne-like identities. Furthermore, we also define hyper-dual vectors by using Pell and Pell-Lucas vectors and discuse hyper-dual angles. This extensionis not only dependent on our understanding of dual numbers, it also highlights the interconnectedness between integer sequences and geometric concepts.

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