On complex Leonardo numbers

Karataş, Adnan

doi:10.7546/nntdm.2022.28.3.458-465

Cited by 13 publications

(7 citation statements)

References 9 publications

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“…Kuhapatanakul and Juthamas [12] extended this study to the generalized Leonardo numbers along with their matrix representation and also in [20] the authors studied the matrix representation of Leonardo numbers. Karatas [10] defined the complex Leonardo numbers and studied their combinatorial properties.…”

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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Section: Introductionmentioning

confidence: 99%

On The Generalized Leonardo Quaternions and Associated Spinors

KUMARI,

PRASAD,

MAHATO

et al. 2024

KgJMat

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“…For other extensions of n , see, e.g., [18,21,22,25]. 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.…”

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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“…where α and β are given Equation 4. Other studies about Leonardo numbers can be listed in [9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions

Dişkaya

Menken

Catarino

2023

Journal of New Theory

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In this paper, we present a new definition, referred to as the Francois sequence, related to the Lucas-like form of the Leonardo sequence. We also introduce the hyperbolic Leonardo and hyperbolic Francois quaternions. Afterward, we derive the Binet-like formulas and their generating functions. Moreover, we provide some binomial sums, Honsberger-like, d’Ocagne-like, Catalan-like, and Cassini-like identities of the hyperbolic Leonardo quaternions and hyperbolic Francois quaternions that allow an understanding of the quaternions' properties and their relation to the Francois sequence and Leonardo sequence. Finally, considering the results presented in this study, we discuss the need for further research in this field.

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On complex Leonardo numbers

Abstract: In this study, we introduce the complex Leonardo numbers and give some of their properties including Binet formula, generating function, Cassini and d’Ocagne’s identities. Also, we calculate summation formulas for complex Leonardo numbers involving complex Fibonacci and Lucas numbers.

Cited by 13 publications

References 9 publications

On The Generalized Leonardo Quaternions and Associated Spinors

On The Generalized Leonardo Quaternions and Associated Spinors

Determinants of Toeplitz–Hessenberg Matrices with Generalized Leonardo Number Entries

On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions

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