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Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions

Year 2023, Volume: 6 Issue: 2, 98 - 101, 01.04.2023

Zafer Ünal

https://doi.org/10.34248/bsengineering.1253548

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

  • Bilgici G. 2014. New generalizations of Fibonacci and Lucas sequences. Appl Math Sci, 8(29): 1429-1437.
  • Bilgici G, Tokeşer Ü, Ünal Z. 2017. Fibonacci and Lucas Sedenions. J Integer Seq, 20: 17.1.8.
  • Cawagas RE. 2004. On the structure and zero divisors of the Cayley-Dickson sedenion algebra. Discuss Math Gen Algebra Appl, 24: 251-265.
  • Clifford WK. 1871. Preliminary sketch of bi-quaternions. Proc Lond Math Soc, 4(1): 381-395.
  • Horadam AF. 1961. A generalized Fibonacci sequence. The American Math Monthly, 68(5): 455-459.
  • Imaeda K, Imaeda M, 2000. Sedenions: algebra and analysis. Appl Math Comput, 115: 77-88.
  • Koshy T. 2001. Fibonacci and lucas numbers with applications. Wiley-Interscience Publication, Quebec, Canada, pp: 77-78.
  • Muskat JB. 1993. Generalized Fibonacci and Lucas sequences and rootfinding methods. Math Comput, 61(203): 365-372.
  • 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.
  • Ünal Z, Tokeşer Ü, Bilgici G. 2017. Some properties of dual Fibonacci and dual Lucas octonions. Adv Appl Clifford Algebras, 27: 1907-1916.
  • Wilcox HJ. 1986. Fibonacci sequences of period n in groups. Fibonacci Quart, 24(4): 356-361.
  • Yayenie O. 2011. A note on generalized Fibonacci sequences. Appl Math Comput, 217(12): 5603-5611.

Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions

Year 2023, Volume: 6 Issue: 2, 98 - 101, 01.04.2023

Zafer Ünal

https://doi.org/10.34248/bsengineering.1253548

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

  • Bilgici G. 2014. New generalizations of Fibonacci and Lucas sequences. Appl Math Sci, 8(29): 1429-1437.
  • Bilgici G, Tokeşer Ü, Ünal Z. 2017. Fibonacci and Lucas Sedenions. J Integer Seq, 20: 17.1.8.
  • Cawagas RE. 2004. On the structure and zero divisors of the Cayley-Dickson sedenion algebra. Discuss Math Gen Algebra Appl, 24: 251-265.
  • Clifford WK. 1871. Preliminary sketch of bi-quaternions. Proc Lond Math Soc, 4(1): 381-395.
  • Horadam AF. 1961. A generalized Fibonacci sequence. The American Math Monthly, 68(5): 455-459.
  • Imaeda K, Imaeda M, 2000. Sedenions: algebra and analysis. Appl Math Comput, 115: 77-88.
  • Koshy T. 2001. Fibonacci and lucas numbers with applications. Wiley-Interscience Publication, Quebec, Canada, pp: 77-78.
  • Muskat JB. 1993. Generalized Fibonacci and Lucas sequences and rootfinding methods. Math Comput, 61(203): 365-372.
  • 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.
  • Ünal Z, Tokeşer Ü, Bilgici G. 2017. Some properties of dual Fibonacci and dual Lucas octonions. Adv Appl Clifford Algebras, 27: 1907-1916.
  • Wilcox HJ. 1986. Fibonacci sequences of period n in groups. Fibonacci Quart, 24(4): 356-361.
  • Yayenie O. 2011. A note on generalized Fibonacci sequences. Appl Math Comput, 217(12): 5603-5611.
There are 12 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Research Articles
Authors

Zafer Ünal 0000-0003-2445-1028

Publication Date April 1, 2023
Submission Date February 20, 2023
Acceptance Date March 11, 2023
Published in Issue Year 2023 Volume: 6 Issue: 2

Cite

APA Ünal, Z. (2023). Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions. Black Sea Journal of Engineering and Science, 6(2), 98-101. https://doi.org/10.34248/bsengineering.1253548
AMA Ünal Z. Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions. BSJ Eng. Sci. April 2023;6(2):98-101. doi:10.34248/bsengineering.1253548
Chicago Ünal, Zafer. “Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions”. Black Sea Journal of Engineering and Science 6, no. 2 (April 2023): 98-101. https://doi.org/10.34248/bsengineering.1253548.
EndNote Ünal Z (April 1, 2023) Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions. Black Sea Journal of Engineering and Science 6 2 98–101.
IEEE Z. Ünal, “Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions”, BSJ Eng. Sci., vol. 6, no. 2, pp. 98–101, 2023, doi: 10.34248/bsengineering.1253548.
ISNAD Ünal, Zafer. “Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions”. Black Sea Journal of Engineering and Science 6/2 (April 2023), 98-101. https://doi.org/10.34248/bsengineering.1253548.
JAMA Ünal Z. Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions. BSJ Eng. Sci. 2023;6:98–101.
MLA Ünal, Zafer. “Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions”. Black Sea Journal of Engineering and Science, vol. 6, no. 2, 2023, pp. 98-101, doi:10.34248/bsengineering.1253548.
Vancouver Ünal Z. Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions. BSJ Eng. Sci. 2023;6(2):98-101.
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