Gaussian Bronze Lucas Numbers

Karaaslan, Nusret

doi:10.35193/bseufbd.1038520

Bilecik Şeyh Edebali Üniversitesi Fen Bilimleri Dergisi

2022

DOI: 10.35193/bseufbd.1038520

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Gaussian Bronze Lucas Numbers

Nusret Karaaslan¹

Abstract: Bu çalışmanın amacı Gauss Bronz Lucas sayı dizisini tanıtmak ve incelemektir. İlk olarak Bronz Lucas sayılarını genişleterek Gauss Bronz Lucas sayılarını tanımladık. Daha sonra bu sayı dizisi için Binet formülü ve üreteç fonksiyonunu bulduk. Ayrıca Gauss Bronz Lucas sayıları ile ilgili bazı toplam formülleri ve matrisleri araştırdık. Son olarak, bu dizinin Binet formülünü dikkate alarak Catalan, Cassini ve d’Ocagne özdeşlikleri gibi bilinen eşitlikleri elde ettik.

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2022

2024

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References 15 publications

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Copper ratio obtained by generalizing the Fibonacci sequence

Özkan,

Akkuş

2024

AIP Advances

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In this study, we define a new generalization of the Fibonacci sequence that gives the copper ratio, and we will call it the copper Fibonacci sequence. In addition, inspired by the copper Fibonacci definition, we also define copper Lucas sequences, and then we give the relationships between the terms of these sequences. We present some properties, such as the Binet formulas, special summation formulas, special generating functions, etc. We find the relationships between the roots of the characteristic equation of these sequences and the general terms of these sequences. What is interesting here is that the relationships obtained from that between the roots of the characteristic equation of these new sequences and the terms of the sequences are satisfied in both roots. In addition, we examine the relationships between these sequences with the classic Fibonacci and Lucas sequences. Moreover, we calculate some identities of these sequences, such as Cassini and Catalan. Then Catalan transformation is applied to these sequences, and their terms are found. Furthermore, we apply Hankel transform to the Catalan transform of these sequences. Besides, we associate the terms of the Hankel transformation of the Catalan copper Fibonacci sequence with the classical Fibonacci numbers and the terms of the Hankel transformation of the Catalan copper Lucas sequence with the terms of the copper Lucas sequence. We present the application of copper Fibonacci and copper Lucas sequences to hyperbolic quaternions. Finally, the terms of the copper Fibonacci and copper Lucas sequences are associated with their hyperbolic quaternion values.

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Copper ratio obtained by generalizing the Fibonacci sequence

Özkan,

Akkuş

2024

AIP Advances

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d-Gaussian Pell polynomials and their matrix representation

Özkan

Uysal

2022

Discrete Math. Algorithm. Appl.

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We introduce a new generalization of Gaussian Pell polynomials. We call it [Formula: see text]-Gaussian Pell polynomials. Then we present the sum, generating functions and Binet formulas of these polynomials. We give the matrix representation of [Formula: see text]-Gaussian Pell polynomials. We introduce the matrix as binary representations according to the Riordan matrix representation. Using the Riordan method, we obtain the factorizations of the Pascal matrix involving the polynomial. In addition, we present the inverse of matrices of the polynomial.

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Gaussian Bronze Lucas Numbers

Cited by 2 publications

References 15 publications

Copper ratio obtained by generalizing the Fibonacci sequence

Copper ratio obtained by generalizing the Fibonacci sequence

d-Gaussian Pell polynomials and their matrix representation

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