An extended framework for bihyperbolic generalized Tribonacci numbers

Yıl 2024, Cilt: 73 Sayı: 3, 765 - 786, 27.09.2024

Nurten Gürses , Zehra İşbilir

https://doi.org/10.31801/cfsuasmas.1378136

Öz

The aim of this article is to identify and analyze a new type special number system which is called bihyperbolic generalized Tribonacci numbers (BGTN for short). For this purpose, we give both classical and several new properties such as; recurrence relation, Binet formula, generating function, exponential generating function, summation formulae, matrix formula, and special determinant equations of BGTN . Also, the system of BGTN is quite a big family and includes several type special cases with respect to initial values and $r,~ s, ~t$ values, we give the subfamilies and special cases of it. In addition to these, we construct some numerical algorithms including recurrence relation and special two types determinant equations related to calculating the terms of this new type special number system. Then, we examine several properties by taking two special cases and including some illustrative numerical examples.

Anahtar Kelimeler

Generalized Tribonacci numbers, bihyperbolic numbers, bihyperbolic generalized Tribonacci numbers, special sequences

Kaynakça

• Adegoke, K., Basic properties of a generalized third order sequence of numbers, ArXiv preprint, (2019), 12 pages. https://ArXiv.org/abs/1906.00788
• Akyiğit, M., Kösal, H. H., Tosun, M., Fibonacci generalized quaternions, Adv. Appl. Clifford Algebr., 24(3) (2014), 631-641. https://doi.org/10.1007/s00006-014-0458-0
• Azak, A. Z., Some new identities with respect to bihyperbolic Fibonacci and Lucas numbers, International Journal of Sciences: Basic and Applied Sciences, 60 (2021), 14-37.
• Bilgin, M., Ersoy, S., Algebraic properties of bihyperbolic numbers, Adv. Appl. Clifford Algebr., 30 (2020), Article number: 13, 17 pages. https://doi.org/10.1007/s00006-019-1036-2
• Bród, D., Szynal-Liana, A., Włoch, I., On a new generalization of bihyperbolic Pell numbers, Annals of the Alexandru Ioan Cuza University-Mathematics, 67(2) (2021), 251-260. https://doi.org/10.47743/anstim.2021.00018
• Bród, D., Szynal-Liana, A., Włoch, I., One-parameter generalization of the bihyperbolic Jacobsthal numbers, Asian-Eur. J. Math., 16(05) (2022), 2350075. https://doi.org/10.1142/S1793557123500754
• Bród, D., Szynal-Liana, A., Włoch, I., Two-parameter generalization of bihyperbolic Jacobsthal numbers, Proyecciones (Antofagasta, Online), 41(3) (2022), 569-578. https://doi.org/10.22199/issn.0717-6279-4071
• Bród, D., Szynal-Liana, A., Włoch, I., Bihyperbolic numbers of the Fibonacci type and their idempotent representation, Comment. Math. Univ. Carolinae, 62(4) (2021), 409-416.
• Bród, D., Szynal-Liana, A., Włoch, I., On some combinatorial properties of bihyperbolic numbers of the Fibonacci type, Math. Methods Appl. Sci., 44(6) (2021), 4607-4615. https://doi.org/10.1002/mma.7054
• Catoni, F., Boccaletti, F., Cannata, R., Catoni, V., Nichelatti, E., Zampetti, P., The Mathematics of Minkowski Space-Time with an Introduction to Commutative Hypercomplex Numbers, Birkhäuser Verlag, Basel, Boston, Berlin, 2008.
• Cerda-Morales, G., On a generalization for Tribonacci quaternions, Mediterr. J. Math., 14 (2017), Article: 239, 12 pages. https://doi.org/10.1007/s00009-017-1042-3
• Cerda-Morales, G., On the third-order Jacobsthal and third-order Jacobsthal-Lucas sequences and their matrix representations, Mediterr. J. Math., 16 (2019), Article: 32, 12 pages. https://doi.org/10.1007/s00009-019-1319-9
• Cerda-Morales, G., A note on modified third-order Jacobsthal numbers, Proyecciones (Antofagasta), 39(2) (2020), 409-420. https://doi.org/10.22199/issn.0717-6279-2020-02-0025
• Cerda-Morales, G., On bicomplex third-order Jacobsthal numbers, Complex Var. Elliptic Equ., 68(1) (2023), 43-56. https://doi.org/10.1080/17476933.2021.1975113
• Cereceda, J. L., Binet’s formula for generalized Tribonacci numbers, International Journal of Mathematical Education in Science and Technology, 46(8) (2015), 1235-1243. https://doi.org/10.1080/0020739X.2015.1031837
• Cereceda, J. L., Determinantal representations for generalized Fibonacci and Tribonacci numbers, Int. J. Contemp. Math. Sci., 9(6) (2014), 269-285. http://dx.doi.org/10.12988/ijcms.2014.4323
• Cockle, J., On systems of algebra involving more than one imaginary; and on equations of the fifth degree, Philosophical Magazine, 35(238) (1849), 434-437.
• Dunlap, R. A., The Golden Ratio and the Fibonacci Numbers, World Scientific, Singapore, 1997.
• Feinberg, M., Fibonacci-Tribonacci, Fibonacci Quart., 1(3) (1963), 71-74.
• Flaut, C., Shpakivskyi, V., On generalized Fibonacci quaternions and Fibonacci-Narayana quaternions, Adv. Appl. Clifford Algebr., 23(3) (2013), 673-688. https://doi.org/10.1007/s00006-013-0388-2
• Günay, H., Taşkara, N., Some properties of Padovan quaternion, Asian-Eur. J. Math., 12(06) (2019), 2040017, 8 pages. https://doi.org/10.1142/S1793557120400173
• Halıcı, S., On Fibonacci quaternions, Adv. Appl. Clifford Algebr., 22(2) (2012), 321-327. https://doi.org/10.1007/s00006-011-0317-1
• Hamilton, W. R., III. On quaternions; or on a new system of imaginaries in algebra, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 25 (1844), 489-495. https://doi.org/10.1080/14786444408644923
• Hamilton, W. R., Lectures on Quaternions, Hodges and Smith, 1853.
• Horadam, A. F., Complex Fibonacci numbers and Fibonacci quaternions, Am. Math. Mon., 70(3) (1963), 289-291. https://doi.org/10.2307/2313129
• Horadam, A. F., Basic properties of a certain generalized sequence of numbers, Fibonacci Quart., 3(3) (1965), 161-176.
• Iyer, M. R., Some results on Fibonacci quaternions, Fibonacci Quart., 7(2) (1969), 201-210.
• İşbilir, Z., Gürses, N., Padovan and Perrin generalized quaternions, Math. Methods Appl. Sci., 45 (2022), 12060-12076. https://doi.org/10.1002/mma.7495
• Jafari, M., Yaylı, Y., Generalized quaternions and rotation in 3-space $E^{3}_{\alpha\beta }$, TWMS J. Pure Appl. Math., 6(2) (2015), 224-232.
• Jafari, M., Yaylı, Y., Generalized quaternions and their algebraic properties, Commun. Fac. Sci. Ank. Series A1, 64(1) (2015), 15-27. https://doi.org/10.1501/Commua1_0000000724
• Kalman, D., Generalized Fibonacci numbers by matrix methods, Fibonacci Quart., 20(1) (1982), 73-76.
• Kızılateş, C., Catarino, P., Tuğlu, N., On the bicomplex generalized Tribonacci quaternions, Mathematics, 7 (2019), Article: 80, 8 pages. https://doi.org/10.3390/math7010080
• Koshy, T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, Inc., New York, 2001.
• Mamagani, A. B., Jafari, M., On properties of generalized quaternion algebra, Journal of Novel Applied Sciences, 2(12) (2013), 683-689.
• Olariu, S., Complex Numbers in n-Dimensions, North-Holland Mathematics Studies, Amsterdam, 2002.
• Pethe, S., Some identities for Tribonacci sequences, Fibonacci Quart., 26(2) (1988), 144-151.
• Pogorui, A. A., Rodriguez-Dagnino, R. M., Rodriguez-Said, R. D., On the set of zeros of bihyperbolic polynomials, Complex Var. Elliptic Equ., 53(7) (2008), 685-690. https://doi.org/10.1080/17476930801973014
• Pottmann, H.,Wallner, J., Computational Line Geometry, Springer-Verlag Berlin Heidelberg, New York, 2001.
• Rochon, D., Shapiro, M., On algebraic properties of bicomplex and hyperbolic numbers, An Univ. Oradea Fasc. Mat., 11(71) (2004), 110.
• Shannon, A. G., Anderson, P. G., Horadam, A. F., Properties of Cordonnier, Perrin and Van der Laan numbers, Int. J. Math. Educ. Sci. Technol., 37(7) (2006), 825-831. https://doi.org/10.1080/00207390600712554
• Shannon, A. G., Horadam, A. F., Some properties of third-order recurrence relations, Fibonacci Quart., 10(2) (1972), 135-146.
• Sloane, N., The Online Encyclopedia of Integer Sequences, 1964, http://oeis.org/.
• Sobczyk, G., The hyperbolic number plane, Coll. Math. J., 26(4) (1995), 268-280. https://doi.org/10.2307/2687027
• Soykan, Y., On generalized third-order Pell numbers, Asian J. Adv. Res. Rep., 6(1) (2019), 1-18.
• Soykan, Y., On generalized Grahaml numbers, J. Adv. Math. Comput. Sci., 35(2) (2020), 42-57. https://doi.org/10.9734/jamcs/2020/v35i230248
• Soykan, Y., Generalized Pell-Padovan numbers, Asian J. Adv. Res. Rep., 11(2) (2020), 8-28. https://doi.org/10.9734/ajarr/2020/v11i230259
• Soykan, Y., A note on binomial transform of the generalized 3-primes sequence, MathLAB Journal, 7(1) (2020), 168-190.
• Soykan, Y., On four special cases of generalized Tribonacci sequence: Tribonacci-Perrin, modified Tribonacci, modified Tribonacci-Lucas and adjusted Tribonacci-Lucas sequences, Journal of Progressive Research in Mathematics, 16(3) (2020), 3056-3084.
• Soykan, Y., On generalized Narayana numbers, Int. J. Adv. Appl. Math. Mech., 7(3) (2020), 43-56.
• Soykan, Y., On generalized reverse 3-primes numbers, Journal of Scientific Research and Reports, 26(6) (2020), 1-20. https://doi.org/10.9734/jsrr/2020/v26i630267
• Soykan, Y., A study on generalized Jacobsthal-Padovan numbers, Earthline Journal of Mathematical Sciences, 4(2) (2020), 227-251. https://doi.org/10.34198/ejms.4220.227251
• Soykan, Y., Summing formulas for generalized Tribonacci numbers, Univers. J. Math. Appl., 3(1) (2020), 1-11. https://doi.org/10.32323/ujma.637876
• Soykan, Y., A study on generalized (r, s, t)-numbers, MathLAB Journal, 7 (2020), 101-129.
• Soykan, Y., On generalized Padovan numbers, Int. J. Adv. Appl. Math., 10(4) (2023), 72-90.
• Szynal-Liana, A., Włoch, I., A study on Fibonacci and Lucas bihypernomials, Discussiones Mathematicae-General Algebra and Applications, 42(2) (2022), 409-423. https://doi.org/10.7151/dmgaa.1399
• Szynal-Liana, A., Włoch, I., Liana, M., On certain bihypernomials related to Pell and Pell-Lucas numbers, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 71(2) (2022), 422-433. https://doi.org/10.31801/cfsuasmas.890932
• Şentürk, T. D., Ünal, Z., 3-parameter generalized quaternions, Computational Methods and Function Theory, 22(3) (2022), 575-608. https://doi.org/10.1007/s40315-022-00451-7
• Taşcı, D., Padovan and Pell-Padovan quaternions, Journal of Science and Arts, 42(1) (2018), 125-132.
• Waddill, M. E., Using matrix techniques to establish properties of a generalized Tribonacci sequence, Applications of Fibonacci Numbers, 4 (1991), 299-308. https://doi.org/10.1007/978-94-011-3586-3_33
• Waddill, M. E., Sacks, L., Another generalized Fibonacci sequence, Fibonacci Quart., 5(3) (1967), 209-222.
• Yaglom, I. M., A Simple Non-Euclidean Geometry and Its Physical Basis, Springer-Verlag, New York, 1979.
• Yalavigi, C. C., Properties of Tribonacci numbers, Fibonacci Quart., 10(3) (1972), 231-246.