Bihyperbolic numbers of the Fibonacci type as tridiagonal  matrix determinants

Dorota Bród; Anetta Szynal-Liana

doi:10.5269/bspm.68492

Dorota Bród Rzeszow University of Technology
Anetta Szynal-Liana Rzeszow University of Technology

Résumé

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.

Téléchargements

Les données sur le téléchargement ne sont pas encore disponible.

Références

Bilgin, M., Ersoy, S., Algebraic properties of bihyperbolic numbers, Adv. Appl. Clifford Algebr. 30:13 (2020). https://doi.org/10.1007/s00006-019-1036-2

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

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

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

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.

Cahill, N. D., Narayan, D. A., Fibonacci and Lucas numbers as tridiagonal matrix determinants, Fibonacci Quart. 42(3) (2004), 216–221.

Catarino, P., Bicomplex k-Pell quaternions, Comput. Methods Funct. Theory, 19 (2019), 65–76. https://doi.org/10.1007/s40315-018-0251-5

Chen, K. W., Horadam sequences and tridiagonal determinants, Symmetry (2020), 12, 1968. https://doi.org/10.3390/sym12121968

Civciv, H., A note on the determinant of five-diagonal matrices with Fibonacci numbers, Int. J. Contemp. Math. Sciences 3(9) (2008), 419–424.

Cockle, J., On a new imaginary in algebra, Lond. Edinb. Dubl. Phil. Mag. 34 (1849), 37–47. https://doi.org/10.1080/14786444908646169

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

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

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

Davis, C., The norm of the Schur product operation, Numer. Math. 4 (1962), 343–344. https://doi.org/10.1007/BF01386329

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

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.

J´ına, 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

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

Kaygısız, K., S¸ahin, A., Determinant and permanent of Hessenberg matrix and Fibonacci type numbers, Gen. Math. Notes 9(2) (2012), 32–41.

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

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

Minc, H., Permanents, Encyclopedia Math. Appl., Vol. 6, Addison–Wesley Publishing Company, London, 1978.

Muir, T., A treatise on the theory of determinants, Dover Publications, 1960.

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

Rochon, D., Shapiro, M., On algebraic properties of bicomplex and hyperbolic numbers, An. Univ. Oradea Fasc. Mat. 11 (2004), 71–110.

Sobczyk, G., The Hyperbolic Number Plane, College Math. J. 26(4) (1995). https://doi.org/10.1080/07468342.1995.11973712

Strang, G., Linear algebra and its applications, Brooks/Cole, 3rd edition, 1988.

Szynal-Liana, A., W loch, I., A study on Fibonacci and Lucas bihypernomials, Discuss. Math. Gen. Algebra Appl. 42(2) (2022), 409–423.

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.

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

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.