Dual-Gaussian Pell and Pell-Lucas numbers
Abstract
In this study, we define a new type of Pell and Pell-Lucas numbers which are called dual-Gaussian Pell and dual-Gaussian Pell-Lucas numbers. We also give the relationship between negadual-Gaussian Pell and Pell-Lucas numbers and dual-complex Pell and Pell-Lucas numbers. Also, some sum ve product properties of Pell and Pell-Lucas numbers are given. Moreover, we obtain the Binet’s formula, generating function, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity and some sum formulas for these new type numbers. Some algebraic proporties of dual-Gaussian Pell and Pell-Lucas numbers are investigated. Futhermore, we give the matrix representation of dual-Gaussian Pell and Pell-Lucas numbers.
Keywords
Dual-Gaussian numbers, Dual-Gaussian Pell numbers, Dual- Gaussian Pell-Lucas numbers
References
- [1] Horadam A.F., Complex Fibonacci Numbers and Fibonacci Quaternions, American Math. Monthly, 70 (1963) 289-291.
- [2] Fjelstad P., Gal S.G., n-dimensional Dual Complex Numbers, Advances in Applied Clifford Algebras, 8(2) (1998) 309-322.
- [3] Clifford W.K., A Preliminary Sketch of Biquaternions, (1873).
- [4] Messelmi F., Dual Complex Numbers and Their Holomorphic Functions, https://hal.archives-ouvertes.fr/hal-01114178. Retrieved January 7, 2022.
- [5] Catarino P., Bicomplex k-Pell Quaternions, Computational Methods and Function Theory, 19 (2019) 65-76.
- [6] Gül K., Dual Bicomplex Horadam Quaternions, Notes on Numbers Theory and Discrete Mathematics, 26 (2020) 187-205.
- [7] Soykan S., On Dual Hyperbolic Generalized Fibonacci Numbers, Indian Journal of Pure and Applied Mathematics, 52 (2021) 62-78.
- [8] Vajda S., Fibonacci and Lucas Numbers and the Golden Section, Ellis Horwood Limited Publ., England, (1989).
- [9] Matsuda G., Kaji S. Ochiai H., Anti-commutative Dual Complex Numbers and 2 Rigid Transformation in Mathematical Progress in Expressive Inage Synthesis I. Springer, (2014). [10] Akar M., Yüce S., Şahin S., On the Dual Hyperbolic Numbers and the Complex Hyperbolic numbers, Journal of Computer Science & Computational Mathematics, 8 (2018) 1-6.
- [11] Majernik V., Multicomponent Number Systems, Acta Physica Polonica A., 90(3) (1996) 491-498.