Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions

Abstract

Fibonacci and Lucas numbers have been the most popular integer sequences since they were defined. These integer sequences have many uses, from nature to computer science, from art to financial analysis. Many researchers have worked on this subject. Sedenions form a 16-dimensional algebra on the field of real numbers. Various systems can be constructed by using the terms of special integer sequences instead of terms in sedenions. In this study, we define dual Fibonacci (DFS) and dual Lucas sedenions (DLS) with the help of Fibonacci and Lucas termed sedenions. Then we calculate some special identities for DFS and DLS such as Vajda's, Catalan's, d'Ocagne's, Cassini's.

Keywords

• Fibonacci and Lucas numbers
• Dual numbers
• Sedenions
• Vajda’s identity
• Binet-like formula

References

1. Bilgici G. 2014. New generalizations of Fibonacci and Lucas sequences. Appl Math Sci, 8(29): 1429-1437.
2. Bilgici G, Tokeşer Ü, Ünal Z. 2017. Fibonacci and Lucas Sedenions. J Integer Seq, 20: 17.1.8.
3. Cawagas RE. 2004. On the structure and zero divisors of the Cayley-Dickson sedenion algebra. Discuss Math Gen Algebra Appl, 24: 251-265.
4. Clifford WK. 1871. Preliminary sketch of bi-quaternions. Proc Lond Math Soc, 4(1): 381-395.
5. Horadam AF. 1961. A generalized Fibonacci sequence. The American Math Monthly, 68(5): 455-459.
6. Imaeda K, Imaeda M, 2000. Sedenions: algebra and analysis. Appl Math Comput, 115: 77-88.
7. Koshy T. 2001. Fibonacci and lucas numbers with applications. Wiley-Interscience Publication, Quebec, Canada, pp: 77-78.
8. Muskat JB. 1993. Generalized Fibonacci and Lucas sequences and rootfinding methods. Math Comput, 61(203): 365-372.

9. Tokeşer Ü, Mert T, Dündar Y. 2022. Some properties and Vajda theorems of split dual Fibonacci and split dual Lucas octonions. AIMS Math, 7(5): 8645-8653.
10. Ünal Z, Tokeşer Ü, Bilgici G. 2017. Some properties of dual Fibonacci and dual Lucas octonions. Adv Appl Clifford Algebras, 27: 1907-1916.
11. Wilcox HJ. 1986. Fibonacci sequences of period n in groups. Fibonacci Quart, 24(4): 356-361.
12. Yayenie O. 2011. A note on generalized Fibonacci sequences. Appl Math Comput, 217(12): 5603-5611.