Gauss Bronz Lucas Sayıları

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.

Keywords

• Bronz Lucas Sayıları
• Gauss Bronz Lucas Sayıları
• Üreteç Fonksiyonu
• Binet Formülü

Gaussian Bronze Lucas Numbers

Abstract

The present work aims to introduce and study the Gaussian Bronze Lucas number sequence. Firstly, we define Gaussian Bronze Lucas numbers by extending the Bronze Lucas numbers. Then, we find the Binet formula and generating function for this number sequence. We also investigate some sum formulas and matrices related to the Gaussian Bronze Lucas numbers. Finally, we obtain some known equalities like Catalan, Cassini and d’Ocagne identities by considering the Binet formula of this sequence.

Keywords

• Bronze Lucas Numbers
• Gaussian Bronze Lucas Numbers
• Generating Function
• Binet Formula

References

1. Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications. John Wiley and Sons Inc., New York, 511-516.
2. Hoggatt, V.E. (1969). Fibonacci and Lucas Numbers. Houghton Mifflin Company, Boston, 2-8.
3. Koshy, T. (2014). Pell and Pell-Lucas Numbers with Applications. Springer, New York, 115-172.
4. Yağmur, T. (2019). New approach to Pell and Pell-Lucas sequences. Kyungpook Mathematical Journal, 59(1), 23-34.
5. Horadam, A.F. (1996). Jacobsthal representation numbers. Fibonacci Quarterly, 34, 40-54.
6. Horadam, A.F. (1963). Complex Fibonacci numbers and Fibonacci quaternions. American Mathematics Monthly, 70, 289-291.
7. Good, I.J. (1993). Complex Fibonacci and Lucas numbers, continued fractions, and the square root of the golden ratio. Fibonacci Quarterly, 31(1), 7-20.
8. Jordan, J.H. (1965). Gaussian Fibonacci and Lucas numbers. Fibonacci Quarterly, 3, 315-318.

9. Berzsenyi, G. (1977). Gaussian Fibonacci numbers. Fibonacci Quarterly, 15(3), 233-236.
10. Halıcı, S., & Öz, S. (2016). On Gaussian Pell and Pell-Lucas numbers. Ordu University Science and Technology Journal, 6(1), 8-18.
11. Aşçı, M., & Gürel, E. (2013). Gaussian Jacobsthal and Gaussian Jacobsthal-Lucas numbers. Ars Combinatoria, 111, 53-63.
12. Shannon, A. G. (2020). Gaussian binomial coefficients. Notes on Number Theory and Discrete Mathematics, 26(1), 225-229.
13. Özkan, E., & Taştan, M. (2020). A new families of Gauss k-Jacobsthal numbers and Gauss k-Jacobsthal-Lucas numbers and their polynomials. Journal of Science and Arts, 4(53), 893-908.
14. Özkan, E., & Taştan, M. (2021). On a new family of Gauss k-Lucas numbers and their polynomials. Asian-European Journal of Mathematics, 14(6), 2150101.
15. Cerda-Morales, G. (2021). Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal-Lucas polynomials and their properties. Asian-European Journal of Mathematics, 14(5), 2150076.
16. Özkan, E., & Kuloğlu, B. (2021). On the new Narayana polynomials, the Gauss Narayana numbers and their polynomials. Asia-European Journal of Mathematics n, 14(6), 2150100.
17. Sloane, N.J.A. (1964). On-line Encyclopedia of Integer Sequences. http://oeis.org/.
18. Akbıyık, M., & Alo, J. (2021). On the third-order Bronze Fibonacci numbers. Mathematics, 9(20), 2606.
19. Yaşar Kartal, M. (2020). Gaussian Bronze Fibonacci numbers. International Journal on Mathematics, Engineering and Natural Sciences, 4(13), 19-25.