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Dual-Gaussian Pell and Pell-Lucas numbers

Year 2022, Volume: 43 Issue: 4, 665 - 671, 27.12.2022

Hasan Gökbaş

https://doi.org/10.17776/csj.1067983

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.
  • [12] Aydın F.T., Dual-complex k-Fibonacci Numbers, Chaos, Solitons & Fractals, 115 (2018) 1-6.
  • [13] Koshy T., Fibonacci and Lucas Numbers with Applications, A Wiley-Intersience Publication, USA, (2001).
  • [14] Koshy T., Pell and Pell-Lucas Numbers with Applications, Springer New York Heidelberg Dordrecht, London, (2014).
  • [15] Halıcı S., Çürük Ş., On Dual k-bicomplex Numbers and Some Identities Including Them, Fundamental Journal of Mathematics and Applications, 3 (2020) 86-93.
  • [16] Azak Z., Güngör M.A., Investigation of Dual-complex Fibonacci, Dual-complex Lucas Numbers and Their Properties, Adv. Appl. Clifford Algebras, 27 (2017) 3083-3096.
  • [17] Aydın F.T., Dual-complex k-Pell Quaternions, Notes on Numbers Theory and Discrete Mathematics, 25 (2019) 111-125.

Year 2022, Volume: 43 Issue: 4, 665 - 671, 27.12.2022

Hasan Gökbaş

https://doi.org/10.17776/csj.1067983

Abstract

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.
  • [12] Aydın F.T., Dual-complex k-Fibonacci Numbers, Chaos, Solitons & Fractals, 115 (2018) 1-6.
  • [13] Koshy T., Fibonacci and Lucas Numbers with Applications, A Wiley-Intersience Publication, USA, (2001).
  • [14] Koshy T., Pell and Pell-Lucas Numbers with Applications, Springer New York Heidelberg Dordrecht, London, (2014).
  • [15] Halıcı S., Çürük Ş., On Dual k-bicomplex Numbers and Some Identities Including Them, Fundamental Journal of Mathematics and Applications, 3 (2020) 86-93.
  • [16] Azak Z., Güngör M.A., Investigation of Dual-complex Fibonacci, Dual-complex Lucas Numbers and Their Properties, Adv. Appl. Clifford Algebras, 27 (2017) 3083-3096.
  • [17] Aydın F.T., Dual-complex k-Pell Quaternions, Notes on Numbers Theory and Discrete Mathematics, 25 (2019) 111-125.
There are 16 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Hasan Gökbaş 0000-0002-3323-8205

Publication Date December 27, 2022
Submission Date February 3, 2022
Acceptance Date October 2, 2022
Published in Issue Year 2022Volume: 43 Issue: 4

Cite

APA Gökbaş, H. (2022). Dual-Gaussian Pell and Pell-Lucas numbers. Cumhuriyet Science Journal, 43(4), 665-671. https://doi.org/10.17776/csj.1067983
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