Tetranacci and Tetranacci-Lucas Quaternions

Main Article Content

Yüksel Soykan

Abstract

The quaternions form a 4-dimensional Cayley-Dickson algebra. In this paper, we introduce the Tetranacci and Tetranacci-Lucas quaternions. Furthermore, we present some properties of these quaternions and derive relationships between them. We present the generating functions, Binet's formulas and sums formulas of these quaternions. Moreover, we give matrix formulation of Tetranacci and Tetranacci-Lucas quaternions.

Keywords:
Tetranacci numbers, quaternions, Tetranacci quaternions, Tetranacci-Lucas quaternions.

Article Details

How to Cite
Soykan, Y. (2019). Tetranacci and Tetranacci-Lucas Quaternions. Asian Research Journal of Mathematics, 15(1), 1-24. https://doi.org/10.9734/arjom/2019/v15i130137
Section
Original Research Article

References

Sloane NJA. The on-line encyclopedia of integer sequences. Available;http://oeis.org/

Hathiwala GS, Shah DV. Binet-type formula for the sequence of tetranacci numbers by alternate methods. Mathematical Journal of Interdisciplinary Sciences 2017;6(1):37-48.

Melham RS. Some Analogs of the Identity F2n+F2n+1= F22n+1. Fibonacci Quarterly. 1999;305-311.

Natividad LR. On solving fibonacci-like sequences of fourth, fth and sixth order. International Journal of Mathematics and Computing. 2013;3(2).

Singh B, Bhadouria P, Sikhwal O, Sisodiya K. A Formula for Tetranacci-Like Sequence, Gen. Math. Notes, 2014;20(2):136-141.

Waddill, ME. Another generalized fibonacci sequence. Fibonacci Quarterly. 1967;5(3):209-227.

Waddill, ME. The tetranacci sequence and generalizations. The Fibonacci Quarterly. 1992;9-20.

Zaveri MN, Patel JK. Binet's formula for the tetranacci sequence. International Journal of Science and Research (IJSR). 2016;5(12):1911-1914.

Dresden GP, Du Z. A simplied binet formula for k-generalized fibonacci numbers. J. Integer Seq. 2014;9.

Howard FT, Saidak F. Zhou's theory of constructing identities. Congress Numer. 2010;200:225-237.

Hankins TL. Sir William Rowan Hamilton, Johns Hopkins University Press, Baltimore; 1980.

Lewis DW. Quaternion algebras and the algebraic legacy of hamilton's quaternions. Irish Math. Soc. Bulletin 2006;57:41-64.

Ward JP. Quaternions and cayley numbers: Algebra and applications. Kluwer Academic Publishers, London; 1997.

Baez J. The octonions, Bull. Amer. Math. Soc. 2002;39(2):145-205.

Horadam AF. Complex fibonacci numbers and bonacci quaternions. Amer. Math. Monthly 1963;70:289-291.

Halici S. On fibonacci Quaternions. Adv. Appl. Cli ord Algebras. 2012;22:321-327.

Cerda-Morales G. On a generalization for tribonacci quaternions. Mediterranean Journal of Mathematics. 2017;14(239):1-12.

Catarino P. The modied pell and modied k-pell quaternions and octonions. Advances in Applied Cliord Algebras 2016;26:577-590.

Halici S, Karatas A. On a generalization for bonacci quaternions. Chaos Solitons and Fractals. 2017;98:178-182.

PolatlıE. A generalization of fibonacci and lucas quaternions. Advances in Applied Clifford Algebras. 2016;26(2):719-730.

Szynal-Liana A, Wloch I. The Pell quaternions and the Pell octonions. Advances in Applied Clifford Algebras. 2016;26(1):435-440.

Tasci D. On k-Jacobsthal and k-Jacobsthal-Lucas Quaternions. Journal of Science and Arts. 2017;3(40):469-476.

Akkus I, Kızılaslan G. On some properties of tribonacci quaternions; 2017. arXiv:1708.05367v1 [math.CO]

Szynal-Liana A, Wloch I. Some properties of generalized tribonacci quaternions. Scientfic Issues, Jan Dlugosz University in Czestochowa, Mathematics. 2017;XXII:73-81.

Tasci D. Padovan and pell-padovan quaternions. Journal of Science and Arts. 2018;1(42):125-132. 5.

Cerda-Morales G. Identities for third order jacobsthal quaternions. Adv. Appl. Clifford Algebras. 2017;27:1043-1053.

Soykan Y. Gaussian generalized tetranacci numbers. Journal of Advances in Mathematics and Computer Science. 2019;31(3):1-21. Article no.JAMCS.48063.

Melham RS, Shannon AG. A generalization of a result of d'ocagne. The Fibonacci Quarterly. 1995;33(2):135-138.