Gaussian Quaternions Including Biperiodic Fibonacci and Lucas Numbers

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

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