Padovan and Pell-Padovan Octonions

Abstract

In this paper, we define the Padovan and Pell-Padovan octonions by using the Padovan and Pell-Padovan numbers. We give the generating functions, Binet's

formulas, sums formulas and some properties for these octonions. We also present the matrix representations of the Padovan and Pell-Padovan octonions.

Keywords

• Padovan numbers,Pell-Padovan numbers,Padovan octonions,Pell-Padovan octonions

References

1. Akkus, I., Ke\c{c}ilioglu, O., \textit{Split Fibonacci and Lucas octonions}, Adv. Appl. Clifford Algebras, \textbf{25(3)}(2015), 517--525.
2. Baez, J., \textit{The octonions}, Bull. Amer. Math. Soc., \textbf{39(2)}(2002), 145--205.
3. Catarino, P., \textit{The modified Pell and the modified $k$-Pell quaternions and octonions}, Adv. Appl. Clifford Algebras, \textbf{26}(2016), 577--590.
4. Cerda-Morales, G., \textit{On a generalization of Tribonacci Quaternions}, Mediterr. J. Math., \textbf{14:239}(2017), 1--12.
5. Cimen, C.B., \.{I}pek, A., \textit{On Pell quaternions and Pell--Lucas quaternions}, Adv. Appl. Clifford Algebras, \textbf{26(1)}(2016), 39--51.
6. Cimen, C.B., \.{I}pek, A., \textit{On Jacobsthal and Jacobsthal--Lucas octonions}, Mediterr. J. Math., \textbf{14(37)}(2017), 13p.
7. Culbert, C., \textit{Cayley-Dickson algebras and loops}, Journal of Generalized Lie Theory and Applications, \textbf{1(1)}(2007), 1--17.
8. Horadam, A.F., \textit{Complex Fibonacci numbers and Fibonacci quaternions}, Am. Math. Month., \textbf{70}(1963), 289--291.

9. Ke\c{c}ilio\u{g}lu, O., Akku\c{s}, I., \textit{The Fibonacci Octonions}, Adv. Appl. Clifford Algebra, \textbf{25}(2015), 151--158.
10. Kritsana, S., \textit{Matrices formula for Padovan and Perrin sequences}, Applied Mathematics Science, \textbf{7(142)}(2013), 7093--7096.
11. Koshy, T., Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001.
12. Koshy, T., \textit{Fibonacci, Lucas and Pell numbers and Pascal's triangle}, Math. Spectrum, \textbf{43}(2011), 125--132.
13. Ser\^{o}dio, R., \textit{On octonionic polynomials}, Adv. Appl. Clifford Algebras, \textbf{17(2)}(2007), 245--258.
14. Tian, Y., \textit{Matrix representations of octonions and their applications}, Adv. Appl. Clifford Algebras, \textbf{10(1)}(2000), 61--90.