Gaussian Quaternions Including Biperiodic Fibonacci and Lucas Numbers

Year 2023, Volume: 6 Issue: 1, 594 - 604, 10.03.2023

Hasan Gökbaş

https://doi.org/10.47495/okufbed.1117644

Abstract

In this study, we define a type of bi-periodic Fibonacci and Lucas numbers which are called bi-periodic Fibonacci and Lucas Gaussian quaternions. We also give the relationship between negabi-periodic Fibonacci and Lucas Gaussian quaternions and bi-periodic Fibonacci and Lucas Gaussian quaternions. Moreover, we obtain the Binet’s formula, generating function, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity, like-Tagiuri’s identity, Honberger’s identity and some formulas for these new type numbers. Some algebraic proporties of bi-periodic Fibonacci and Lucas Gaussian quaternions which are connected between Gaussian quaternions and bi-periodic Fibonacci and Lucas numbers are investigated.

Keywords

Quaternion, Gaussian quaternion, bi-periodic Fibonacci and Lucas numbers

Biperiodic Fibonacci ve Lucas Sayılarını İçeren Gaussian Quaternionlar

Abstract

Bu çalışmada, bi-periodic Fibonacci ve Lucas sayılarının, bi-periodic Fibonacci ve Lucas Gaussian quaternionlar olarak isimlendirilen yeni bir tipi tanımlanmıştır. Çalışma içerisinde, negabi-periodic Fibonacci ve Lucas Gaussian quaternionlarla bi-periodic Fibonacci ve Lucas Gaussian quaternionlar arasındaki ilişkiden de bahsedilmiştir. Ayrıca, bu sayılar için Binet’s formülü, dizinin genelleştirme fonksiyonu, d’Ocagne’s eşitliği, Catalan’s eşitliği, Cassini’s eşitliği, , like-Tagiuri’s eşitliği, Honberger’s eşitliği ve bazı toplam formülleri verilmiştir. Bi-periodic Fibonacci ve Lucas Gaussian quaternionların bazı cebirsel özellikleri ele alınmıştır.

Keywords

Quaternion, Gaussian quaternion, Bi-periodic Fibonacci ve Lucas sayıları

References

• Adler SL., Quaternionic quantum mechanics and quantum fields. New York Oxford Univ. Press; 1994.
• Baez J., The Octonians. Bull. Amer. Math. Soc. 2001; 145(39): 145-205.
• Bilgici G., Two generalizations of Lucas sequence. Applied Mathematics and Computation 2014; 245: 526-538.
• Çimen CB., İpek A., On Pell quaternions and Pell-Lucas quaternions. Adv. Appl. Clifford Algebras 2016; 26(1): 39–51.
• Edson M., Yayenie O., A new generalization of Fibonacci sequences and the extended Binet’s formula. INTEGERS Electron. J. Comb. Number Theor 2009; 9: 639-654.
• Falcon S., Plaza A., On the Fibonacci k-numbers. Chaos, Solitions &Fractals 2007; 32(5): 1615-1624.
• George AH., Some formula for the Fibonacci sequence with generalization. Fibonacci Quart 1969; 7: 113-130.
• Gökbaş H., A note BiGaussian Pell and Pell-Lucas numbers. Journal of Science and Arts 2021; 3(56): 669-680.
• Harman CJ., Complex Fibonacci numbers. The Fibonacci Quart 1981; 19(1): 82-86.
• Horadam AF., A generalized Fibonacci sequence. Math. Mag. 1961; 68: 455-459.
• Horadam AF., Complex Fibonacci numbers and Fibonacci quaternions. Amer. Math. Montly 1963; 70: 289-291.
• Pethe SP., Phadte CN., A generalization of the Fibonacci sequence. Applications of Fibonacci numbers 1992; 5: 465-472.
• Pond JC., Generalized Fibonacci summations. Fibonacci Quart 1968; 6: 97-108.
• Ramirez J., Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions. An. Şt. Univ. Ovidius Constanta 2015; 23(2): 201-212.
• Ward JP., Quaternions and Cayley numbers. Algebra and Applications. Kluwer Academic Publishers, London, 1997.