On Pell and Pell-Lucas Generalized Octonions

In this study, we gave a generalization on Pell and Pell-Lucas octonions over the algebra O(a, b, c) where a, b and c are real numbers. For these number sequences, we obtain Binet formulas and gave some wellknown identities such as Catalan’s identity, Cassini’s identity and d’Ocagne’s identity.

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[1] Akkus, I., Kecilioglu, O., Split Fibonacci and Lucas Octonions, Adv. Appl. Clifford Algebras, 25(2015), 517–525.

[2] Akyiğit, M., Kösal, H.H., Tosun, M., Fibonacci Generalized Quaternions, Adv. Appl. Clifford Algebras, 24(2014), 631–641.

[3] Catarino, P., The Modified Pell and The Modified k-Pell Quaternions and Octonions, Adv. Appl. Clifford Algebras, 26(2)(2016), 577–590.

[4] Cimen, C.B., İpek, A., On Pell Quaternions and Pell-Lucas Quaternions, Adv. Appl. Clifford Algebras, 26(2016), 39–51.

[5] Flaut, C., Stefanescu, M., Some Equations over Generalized Quaternion and Octonion Division Algebras, Bull. Math. Soc. Sci. Math. Roum., 52(100)(4)(2009), 427–439.

[6] Halici, S., On Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 22(2012), 321–327.

[7] Horadam, A.F., Complex Fibonacci Numbers and Fibonacci Quaternions, Amer. Math. Monthly, 70(1963), 289–291.

[8] Horadam, A.F., Quaternion Recurrence Relations, Ulam Quarterly, 2(1993), 23–33.

[9] Iakin, A.L., Generalized Quaternions of Higher Order, Fibonacci Quart, 15(1977), 343–346.

[10] Iyer, M.R., A Note on Fibonacci Quaternions, Fibonacci Quart, 3(1969), 225–229.

[11] Keçilioğlu, O., Akkus, I., The Fibonacci Octonions, Adv. Appl. Clifford Algebras, 25(2015), 151–158.

[12] Koshy, T., Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, Canada, 2001.

[13] Koshy, T., Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.

[14] Ramirez, J.L., Some Combinatorial Properties of the k-Fibonacci and the k-Lucas Quaternions, An. St. Univ. Ovidius Constanta, 23(2)(2015), 201–212.

[15] Savin, D., Some Properties of Fibonacci numbers, Fibonacci Octonions, and generalized Fibonacci-Lucas Octonions, Adv. Difference Equ., 2015.1(2015), 298.

[16] Swamy, M.N.S., On Generalized Fibonacci Quaternions, The Fib. Quarterly, 5(1973), 547–550.

[17] Szynal-Liana, A., Wloch, I., The Pell Quaternions and The Pell Octonions, Adv. Appl. Clifford Algebras, 26(2016), 435–440

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