Bihyperbolic numbers of the Fibonacci type as tridiagonal matrix determinants
DOI:
https://doi.org/10.5269/bspm.68492Resumen
Bihyperbolic numbers are extension of hyperbolic numbers to four dimensions. In this paper, we construct a family of tridiagonal matrices which determinants (continuants) and permanents can represent the sequences of bihyperbolic numbers of the Fibonacci type.
Referencias
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21. Kilic, E., Ta¸sci, D., On the permanents of some tridiagonal matrices with applications to the Fibonacci and Lucas numbers, Rocky Mountain J. Math. 37(6) (2007), 1953-1969. http://dx.doi.org/10.1216/rmjm/1199649832
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25. Rochon, D., Shapiro, M., On algebraic properties of bicomplex and hyperbolic numbers, An. Univ. Oradea Fasc. Mat. 11 (2004), 71–110.
26. Sobczyk, G., The Hyperbolic Number Plane, College Math. J. 26(4) (1995). https://doi.org/10.1080/07468342.1995.11973712
27. Strang, G., Linear algebra and its applications, Brooks/Cole, 3rd edition, 1988.
28. Szynal-Liana, A., W loch, I., A study on Fibonacci and Lucas bihypernomials, Discuss. Math. Gen. Algebra Appl. 42(2) (2022), 409–423.
29. Szynal-Liana, A., W loch, I., Liana, M., On certain bihypernomials related to Pell and Pell-Lucas numbers, Commun. Fac. Sci. Univ. Ank. S´er. A1 Math. Stat., vol. 71, Issue 2 (2022), 422–433.
30. Trojovsky, P., On a sequence of tridiagonal matrices whose determinants are Fibonacci numbers Fn+1, Int. J. Pure Appl. Math. 102(3) (2015), 527–532. http://dx.doi.org/10.12732/ijpam.v102i3.10
31. Ya¸sar, M., Bozkurt, D., Another proof of Pell identities by using the determinant of tridiagonal matrix, Appl. Math. Comput. 218(10) (2012), 6067–6071. https://doi.org/10.1016/j.amc.2011.11.089.
2. Brod, D., Szynal-Liana, A., W loch, I., Bihyperbolic numbers of the Fibonacci type and their idempotent representation, Comment. Math. Univ. Carolin. 62,4 (2021), 409-416. http://dx.doi.org/10.14712/1213-7243.2021.033
3. Brod, D., Szynal-Liana, A., W loch, I., On some combinatorial properties of bihyperbolic numbers of the Fibonacci type, Math. Methods Appl. Sci. 44 (2021), 4607–4615. https://doi.org/10.1002/mma.7054
4. Cahill, N. D., D’Errico, J. R., Narayan, D. A., Narayan, J. Y., Fibonacci determinants, The College Mathematics Journal 33:3 (2002), 221–225. https://doi.org/10.1080/07468342.2002.11921945
5. Cahill, N. D., D’Errico, J. R., Spence, J. P., Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart., 41(1) (2003), 13–19.
6. Cahill, N. D., Narayan, D. A., Fibonacci and Lucas numbers as tridiagonal matrix determinants, Fibonacci Quart. 42(3) (2004), 216–221.
7. Catarino, P., Bicomplex k-Pell quaternions, Comput. Methods Funct. Theory, 19 (2019), 65–76. https://doi.org/10.1007/s40315-018-0251-5
8. Chen, K. W., Horadam sequences and tridiagonal determinants, Symmetry (2020), 12, 1968. https://doi.org/10.3390/sym12121968
9. Civciv, H., A note on the determinant of five-diagonal matrices with Fibonacci numbers, Int. J. Contemp. Math. Sciences 3(9) (2008), 419–424.
10. Cockle, J., On a new imaginary in algebra, Lond. Edinb. Dubl. Phil. Mag. 34 (1849), 37–47. https://doi.org/10.1080/14786444908646169
11. Cockle, J., On certain functions resembling quaternions, and on a new imaginary in algebra, Lond. Edinb. Dubl. Phil. Mag. 33 (1848), 435–439. https://doi.org/10.1080/14786444808646139
12. Cockle, J., On impossible equations, on impossible quantities, and on tessarines, Lond. Edinb. Dubl. Phil. Mag. 37 (1850), 281–283. https://doi.org/10.1080/14786445008646598
13. Cockle, J., On the symbols of algebra, and on the theory of tesarines, Lond. Edinb. Dubl. Phil. Mag. 34 (1849), 406–410. https://doi.org/10.1080/14786444908646257
14. Davis, C., The norm of the Schur product operation, Numer. Math. 4 (1962), 343–344. https://doi.org/10.1007/BF01386329
15. Gulec, H. H., Permanents and determinants of tridiagonal matrices with (s, t)-Pell Lucas numbers, Int. J. Math. Anal. (Ruse) 11(23) (2017), 1117–1122. https://doi.org/10.12988/ijma.2017.7690
16. Ipek, A., On the determinants of pentadiagonal matrices with the classical Fibonacci, generalized Fibonacci and Lucas numbers, Eurasian Math. J. 2.(2) (2011), 60–74.
17. J´Ä±na, J., Trojovsky, P., On permanents of some tridiagonal matrices connected with Fibonacci numbers, Int. J. Pure Appl. Math. 97(1) (2014), 79–87. http://dx.doi.org/10.12732/ijpam.v97i1.8
18. Kasempin, P., Vipismakul, W., Kaewsuy, A., Tridiagonal matrices with permanent values equal to k-Jacobsthal sequence, Asian Journal of Applied Sciences 8(5) (2020), 269–274. https://doi.org/10.24203/ajas.v8i5.6348
19. Kaygısız, K., S¸ahin, A., Determinant and permanent of Hessenberg matrix and Fibonacci type numbers, Gen. Math. Notes 9(2) (2012), 32–41.
20. Kaygısız, K., S¸ahin, A., Determinantal and permanental representations of Fibonacci type numbers and polynomials, Rocky Mountain J. Math. 46(1) (2016), 227–242. https://doi.org/10.1216/RMJ-2016-46-1-227
21. Kilic, E., Ta¸sci, D., On the permanents of some tridiagonal matrices with applications to the Fibonacci and Lucas numbers, Rocky Mountain J. Math. 37(6) (2007), 1953-1969. http://dx.doi.org/10.1216/rmjm/1199649832
22. Minc, H., Permanents, Encyclopedia Math. Appl., Vol. 6, Addison–Wesley Publishing Company, London, 1978.
23. Muir, T., A treatise on the theory of determinants, Dover Publications, 1960.
24. Qi, F., Kızılate¸s, C., Du, W. S., A closed formula for the Horadam polynomials in terms of a tridiagonal determinant, Symmetry (2019), 11(6), 782. https://doi.org/10.3390/sym11060782
25. Rochon, D., Shapiro, M., On algebraic properties of bicomplex and hyperbolic numbers, An. Univ. Oradea Fasc. Mat. 11 (2004), 71–110.
26. Sobczyk, G., The Hyperbolic Number Plane, College Math. J. 26(4) (1995). https://doi.org/10.1080/07468342.1995.11973712
27. Strang, G., Linear algebra and its applications, Brooks/Cole, 3rd edition, 1988.
28. Szynal-Liana, A., W loch, I., A study on Fibonacci and Lucas bihypernomials, Discuss. Math. Gen. Algebra Appl. 42(2) (2022), 409–423.
29. Szynal-Liana, A., W loch, I., Liana, M., On certain bihypernomials related to Pell and Pell-Lucas numbers, Commun. Fac. Sci. Univ. Ank. S´er. A1 Math. Stat., vol. 71, Issue 2 (2022), 422–433.
30. Trojovsky, P., On a sequence of tridiagonal matrices whose determinants are Fibonacci numbers Fn+1, Int. J. Pure Appl. Math. 102(3) (2015), 527–532. http://dx.doi.org/10.12732/ijpam.v102i3.10
31. Ya¸sar, M., Bozkurt, D., Another proof of Pell identities by using the determinant of tridiagonal matrix, Appl. Math. Comput. 218(10) (2012), 6067–6071. https://doi.org/10.1016/j.amc.2011.11.089.
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2025-03-18
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Cómo citar
Bród, D., & Szynal-Liana, A. (2025). Bihyperbolic numbers of the Fibonacci type as tridiagonal matrix determinants. Boletim Da Sociedade Paranaense De Matemática, 43, 1-7. https://doi.org/10.5269/bspm.68492
