On some combinatorial properties of bihyperbolic numbers of the Fibonacci type

“…On the other hand, a bihyperbolic number ζ = x 0 + j 1 x 1 + j 2 x 2 + j 3 x 3 has three conjugates such that [6]. Considering these conjugates, the hyperbolic valued modulus is introduced [9]. It is defined as…”

Section: Preliminariesmentioning

confidence: 99%

“…Apart from all these, detailed surveys on the algebraic [13], geometric and topological [14], and combinatorial properties [8,9] of bihyperbolic numbers were given. However, bihyperbolic modules and topological bihyperbolic modules have not investigated yet.…”

Section: Introductionmentioning

confidence: 99%

Topological Bihyperbolic Modules

Bilgin

Ersoy

2021

Communications in Advanced Mathematical Sciences

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The aim of this article is introducing and researching hyperbolic modules, bihyperbolic modules, topological hyperbolic modules and topological bihyperbolic modules. In this regard, we define balanced, convex and absorbing sets in hyperbolic and bihyperbolic modules. In particular, we investigate convex sets in hyperbolic numbers set (it is a hyperbolic module over itself) by considering the isomorphic relation of this set with 2−dimensional Minkowski space. Moreover, bihyperbolic numbers set is a bihyperbolic module over itself, too. So, we define convex sets in this module by considering hypersurfaces of 4−dimensional semi Euclidean space that are isomorphic to some subsets of bihyperbolic numbers set. We also study the interior and closure of some special sets and neighbourhoods of the unit element of the module in the introduced topological bihyperbolic modules. In the light of obtained results, new relationships are presented for idempotent representations in topological bihyperbolic modules.

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“…They investigated dual hyperbolic number and hyperbolic complex number valued functions. Brod (Brod, Szynal-Liana, Wloch, 2020) et al formulated any bihyperbolic number by 𝑤 = 𝑥 1 + 𝑗𝑥 2 + 𝑗 1 𝑥 3 + 𝑗 2 𝑥 4 . Addition, substraction and multiplication of bihyperbolic numbers and was defined as…”

Section: Introductionmentioning

confidence: 99%

Gaussian-bihyperbolic Numbers Containing Pell and Pell-Lucas Numbers

Gökbaş

2023

Journal of Advanced Research in Natural and Applied Sciences

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In this study, we define a new type of Pell and Pell-Lucas numbers which are called Gaussian-hyperbolic Pell and Pell-Lucas numbers. We also give negaGaussian-hyperbolic Pell and Pell-Lucas numbers. More-over, we obtain the Binet’s formula, generating function formula, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity and some sum formulas for these new type numbers. Some algebraic proporties of Gaussian-hyperbolic Pell and Pell-Lucas numbers are investigated. Futhermore, we give the matrix repre-sentation of Gaussian-hyperbolic Pell and Pell-Lucas number.

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On some combinatorial properties of bihyperbolic numbers of the Fibonacci type

Abstract: In this paper, we give some properties of the bihyperbolic Fibonacci, Jacobsthal and Pell numbers, among others the Binet formula, Catalan, Cassini, and d'Ocagne identities. Moreover, we present the generating function and summation formulas for these numbers.

Cited by 10 publications

References 8 publications

On certain bihypernomials related to Pell and Pell-Lucas numbers

On certain bihypernomials related to Pell and Pell-Lucas numbers

Topological Bihyperbolic Modules

Gaussian-bihyperbolic Numbers Containing Pell and Pell-Lucas Numbers

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