Pauli–Leonardo quaternions

İşbilir, Zehra; Akyiğit, Mahmut; Tosun, Murat

doi:10.7546/nntdm.2023.29.1.1-16

Cited by 6 publications

(4 citation statements)

References 28 publications

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“…In literature, some authors have been working on Pauli coefficients associated with a numerical sequence, for example. With this in mind, Isbilir et al (2023) originated the Pauli-Leonardo numbers, presenting their quaternions. Additionally, Isbilir et al (2023) investigated these numbers based on the relationships between Pauli-Fibonacci and Pauli-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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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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“…İşbilir et al [23] investigated the Pauli-Leonardo quaternions. Some recent developments on Leonardo numbers, their generalizations and interesting properties can be seen in [3,5,[14][15][16][17][18][19]22].…”

Section: Introductionmentioning

confidence: 99%

On The Generalized Leonardo Quaternions and Associated Spinors

KUMARI,

PRASAD,

MAHATO

et al. 2024

KgJMat

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In this paper, we introduce and study a new family of sequences called the generalized Leonardo spinors by deﬁning a linear correspondence between the generalized Leonardo quaternions and spinors. We start with deﬁning the generalized Leonardo quaternions and then present their some important properties such as Binet type formula, Catalan’s identity, d’Ocagne’s identity, series sums, etc. We give some interrelations of these quaternions with the Fibonacci and Lucas quaternions. Then, we present the generating functions, sum formulae, various well-known identities, etc. for the Leonardo spinors and show their connection with the Fibonacci and Lucas spinors.

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“…Complex Leonardo numbers were considered in [1,15,16], where various properties including recurrences and explicit formulas were shown. Further extensions in terms of hybrid numbers with Leonardo [1] or complex Leonardo [15] coefficients or in terms of the quaternions [14] or octonians [28] have subsequently been studied. Here, we consider some new combinatorial aspects of the generalized Leonardo numbers u n as it pertains to their occurrence in certain Toeplitz-Hessenberg matrices.…”

Section: Introductionmentioning

confidence: 99%

Determinants of Toeplitz–Hessenberg Matrices with Generalized Leonardo Number Entries

Goy,

Shattuck

2024

Annales Mathematicae Silesianae

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Let un = un (k) denote the generalized Leonardo number defined recursively by un = un− 1 + un− 2 + k for n ≥ 2, where u 0 = u 1 = 1. Terms of the sequence un (1) are referred to simply as Leonardo numbers. In this paper, we find expressions for the determinants of several Toeplitz–Hessenberg matrices having generalized Leonardo number entries. These results are obtained as special cases of more general formulas for the generating function of the corresponding sequence of determinants. Special attention is paid to the cases 1 ≤ k ≤ 7, where several connections are made to entries in the On-Line Encyclopedia of Integer Sequences. By Trudi’s formula, one obtains equivalent multi-sum identities involving sums of products of generalized Leonardo numbers. Finally, in the case k = 1, we also provide combinatorial proofs of the determinant formulas, where we make extensive use of sign-changing involutions on the related structures.

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Pauli–Leonardo quaternions

Cited by 6 publications

References 28 publications

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

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

On The Generalized Leonardo Quaternions and Associated Spinors

Determinants of Toeplitz–Hessenberg Matrices with Generalized Leonardo Number Entries

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