The linear algebra of a generalized Tribonacci matrix

Abstract

In this paper, we consider a generalization of a regular Tribonacci matrix for two variables and show that it can be factorized by some special matrices. We produce several new interesting identities and find an explicit formula for the inverse and k−th power. We also give a relation between the matrix and a matrix exponential of a special matrix.

Keywords

• Tribonacci sequence
• factorization of matrices
• matrix exponential
• matrix inverse

References

1. Bayat, M., Teimoori, H., The linear algebra of the generalized Pascal functional matrix, Linear Algebra Appl., 295 (1999), 81–89. https://dx.doi.org/10.1016/S0024-3795(99)00062-2
2. Bayat, M., Teimoori, H., Pascal k−eliminated functional matrix and its property, Linear Algebra Appl., 308 (1–3) (2000), 65–75. https://dx.doi.org/10.1016/S0024-3795(99)00266-9
3. Call, G. S., Velleman, D. J., Pascal matrices, Amer. Math. Monthly, 100 (1993), 372–376. https://doi.org/10.1080/00029890.1993.11990415
4. Catalani, M., Identities for Tribonacci-related sequences, arXiv:math/0209179 [math.CO]. https://doi.org/10.48550/arXiv.math/0209179
5. Choi, E., Modular Tribonacci numbers by matrix method, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math., 20 (2013), 207–221. https://dx.doi.org/10.7468/jksmeb.2013.20.3.207
6. Devbhadra, S. V., Some Tribonacci identities, Math. Today, 27 (2011), 1–9.
7. Edelman, A., Strang, G., Pascal matrices, Amer. Math. Monthly, 111 (3) (2004), 189–197. https://dx.doi.org/10.1080/00029890.2004.11920065
8. Falcon, S., Plaza, A., On the Fibonacci k−numbers, Chaos, Solitons & Fractals, 32 (2007), 1615–1624. https://dx.doi.org/10.1016/j.chaos.2006.09.022

9. Falcon, S., On the k−Lucas Numbers, International Journal of Contemporary Mathematical Sciences, 6 (21) (2011), 1039–1050.
10. Feinberg, M., Fibonacci-Tribonacci, Fibonacci Quart., 1 (1963), 71–74.
11. Horadam, A. F., Basic properties of a certain generalized sequence of numbers, Fibonacci Quart., 3 (1965), 161–176.
12. Horadam, A. F., Special properties of the sequence $W_n(a, b; p, q)$, Fibonacci Quart., 5 (5) (1967), 424–434.
13. Horadam, A. F., Jacobsthal representation numbers, Fibonacci Quart., 34 (1) (1996), 40–53.
14. Horn, R. A., Johnson, C. R., Matrix Analysis, Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne, Sydney, Second Edition, 2013.
15. Howard, F. T., A Tribonacci identity, Fibonacci Quart., 39 (2001), 352–357.
16. Jakubczyk, Z., Sums of Squares of Tribonacci Numbers, Advanced problems and solutions edited by Florian Luca, The Fibonacci Quarterly, August (2013), 4–5.
17. Kalman, D., Mena, R., The Fibonacci numbers-exposed, Math. Mag., 76 (3) (2003), 167–181. https://dx.doi.org/10.1080/0025570X.2003.11953176
18. Kızılateş, C., Terzioğlu, N., On r−min and r−max matrices, Journal of Applied Mathematics and Computing, (2022), 1–30. https://dx.doi.org/10.1007/s12190-022-01717-y
19. Kilic, E., Arikan, T., Studying new generalizations of Max-Min matrices with a novel approach, Turkish Journal of Mathematics, 43 (4) (2019), 2010–2024.
20. Pethe, S., Some identities for Tribonacci sequences, Fibonacci Quart., 26 (1988), 144–151.
21. Piezas, T., A tale of four constants, https://sites.google.com/site/tpiezas/0012.
22. Scott, A., Delaney, T., Hoggatt J. R., V., The Tribonacci sequence, Fibonacci Quart., 15 (1977), 193–200.
23. Spickerman, W., Binet’s formula for the Tribonacci sequence, Fibonacci Quart., 20 (1982), 118–120.
24. Williamson, R., Trotter, H., Multivariable Mathematics, second edition, Prentice-Hall, 1979.
25. Yalavigi, C. C., Properties of Tribonacci numbers, Fibonacci Quart., 10 (3) (1972), 231–246.
26. Yaying, T., Hazarika, B., On sequence spaces defined by the domain of a regular Tribonacci matrix, Mathematica Slovaca, 70 (3) (2020), 697–706. https://dx.doi.org/10.1515/ms-2017-0383
27. Zhang, Z., The linear algebra of the generalized Pascal matrix, Linear Algebra Appl., 250 (1997), 51–60. https://dx.doi.org/10.1016/0024-3795(95)00452-1