Ordered Leonardo Quadruple Numbers

Nurkan, Semra Kaya; Güven, İlkay Arslan

doi:10.3390/sym15010149

Cited by 9 publications

(7 citation statements)

References 28 publications

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“…Taking k = 1 gives the analogous relations for the Leonardo sequence with the dual-quaternions coefficients. Hence, we can say that our main results presented here generalize the paper [13]. These results can trigger further research on the subjects of the Leonardo sequence and the dual quaternions.…”

Section: Discussionsupporting

confidence: 83%

“…Although the Fibonacci, Lucas, and Leonardo sequences are closely related, they exhibit distinct characteristic properties. Several different properties and generalizations of the Leonardo sequence were previously studied by various researchers [3][4][5][6][7][8][9][10][11][12][13][14][15]. Recently, a one-parameter generalized Leonardo sequence has been defined as non-homogeneous recursively by…”

Section: Introductionmentioning

confidence: 99%

“…Considering all these details, a natural question is whether the paper [13] can be generalized. In this study, we aim to determine the dual quaternions with the k−generalized Leonardo sequence.…”

Section: Introductionmentioning

confidence: 99%

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On Dual Quaternions with $k-$Generalized Leonardo Components

YILMAZ,

SAÇLI

2023

Journal of New Theory

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In this paper, we define a one-parameter generalization of Leonardo dual quaternions, namely $k-$generalized Leonardo-like dual quaternions. We introduce the properties of $k$-generalized Leonardo-like dual quaternions, including relations with Leonardo, Fibonacci, and Lucas dual quaternions. We investigate their characteristic relations, involving the Binet-like formula, the generating function, the summation formula, Catalan-like, Cassini-like, d'Ocagne-like, Tagiuri-like, and Hornsberger-like identities. The crucial part of the present paper is that one can reduce the calculations of Leonardo-like dual quaternions by considering $k$. For $k=1$, these results are generalizations of the ones for ordered Leonardo quadruple numbers. Finally, we discuss the need for further research.

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Section: Discussionsupporting

confidence: 83%

Section: Introductionmentioning

confidence: 99%

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On Dual Quaternions with $k-$Generalized Leonardo Components

YILMAZ,

SAÇLI

2023

Journal of New Theory

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“…Regarding the k-Leonardo numbers, Prasad et al (2023) introduced new families of generalized k-Leonardo numbers, as well as Leonardo Gaussians, bringing new properties, generating functions, and identities. Nurkan and Guven (2023) introduce a sequence of quadruple numbers derived from the Leonardo numbers, called ordered Leonardo quadruple numbers. Properties involving these numbers were developed, relating them to Leonardo, Fibonacci, and Lucas numbers.…”

Section: A State Of Art On the Leonardo Sequence: Panorama Of Current...mentioning

confidence: 99%

State of the art on the Leonardo sequence: An evolutionary study of the epistemic-mathematical field

Mangueira,

Alves,

Catarino

et al. 2024

PEDAGOGICAL RES

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This work is a segment of an ongoing doctoral research in Brazil. The Leonardo numbers and the Leonardo sequence have gained attention from mathematicians and the academic community. Despite being a relatively new sequence within mathematical literature, its discussion has intensified over the past five years, giving rise to other branches, with contributions and associations to other topics in mathematics. Thus, the aim of this study was to construct and present the state of the art of the Leonardo sequence, considering its historical aspects and highlighting works on its evolutionary process in the epistemic-mathematical field, regarding its generalization, complexification, hyper complexification, and combinatorial model during the last five years (2019-2023). The methodology used was a bibliographic study, where the state of the art was carried out through the mapping of publications on the subject. Twenty-four research works related to the key descriptors “Leonardo sequence”, “Leonardo numbers”, “complexification”, “generalization”, “hybrids”, and “combinatorial model” were found, cataloged, and discussed. From the analysis of these studies, it is noted that its development in pure mathematics has advanced to other branches and discoveries, and that, albeit timidly, research on the subject has emerged directed towards the field of education, especially in the initial teacher training and, particularly, in Brazil.

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“…For the properties of Leonardo numbers and related studies, we refer to [11][12][13][14][15][16], and for the history of Leonardo sequences, see [A001595] in the On-Line Encyclopedia of Integer Sequences [17].…”

Section: Introductionmentioning

confidence: 99%

A New Class of Leonardo Hybrid Numbers and Some Remarks on Leonardo Quaternions over Finite Fields

Tan,

Savin,

Yılmaz

2023

Mathematics

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In this paper, we present a new class of Leonardo hybrid numbers that incorporate quantum integers into their components. This advancement presents a broader generalization of the q-Leonardo hybrid numbers. We explore some fundamental properties associated with these numbers. Moreover, we study special Leonardo quaternions over finite fields. In particular, we determine the Leonardo quaternions that are zero divisors or invertible elements in the quaternion algebra over the finite field Zp for special values of prime integer p.

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Ordered Leonardo Quadruple Numbers

Cited by 9 publications

References 28 publications

On Dual Quaternions with $k-$Generalized Leonardo Components

On Dual Quaternions with $k-$Generalized Leonardo Components

State of the art on the Leonardo sequence: An evolutionary study of the epistemic-mathematical field

A New Class of Leonardo Hybrid Numbers and Some Remarks on Leonardo Quaternions over Finite Fields

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