An extended framework for bihyperbolic generalized Tribonacci numbers

Abstract

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.

Keywords

• Generalized Tribonacci numbers
• bihyperbolic numbers
• bihyperbolic generalized Tribonacci numbers
• special sequences

References

1. Adegoke, K., Basic properties of a generalized third order sequence of numbers, ArXiv preprint, (2019), 12 pages. https://ArXiv.org/abs/1906.00788
2. 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
3. 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.
4. 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
5. 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
6. 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
7. 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
8. 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.

9. 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
10. 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.
11. 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
12. 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
13. 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
14. 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
15. 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
16. 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
17. 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.
18. Dunlap, R. A., The Golden Ratio and the Fibonacci Numbers, World Scientific, Singapore, 1997.
19. Feinberg, M., Fibonacci-Tribonacci, Fibonacci Quart., 1(3) (1963), 71-74.
20. 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
21. 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
22. Halıcı, S., On Fibonacci quaternions, Adv. Appl. Clifford Algebr., 22(2) (2012), 321-327. https://doi.org/10.1007/s00006-011-0317-1
23. 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
24. Hamilton, W. R., Lectures on Quaternions, Hodges and Smith, 1853.
25. Horadam, A. F., Complex Fibonacci numbers and Fibonacci quaternions, Am. Math. Mon., 70(3) (1963), 289-291. https://doi.org/10.2307/2313129
26. Horadam, A. F., Basic properties of a certain generalized sequence of numbers, Fibonacci Quart., 3(3) (1965), 161-176.
27. Iyer, M. R., Some results on Fibonacci quaternions, Fibonacci Quart., 7(2) (1969), 201-210.
28. İşbilir, Z., Gürses, N., Padovan and Perrin generalized quaternions, Math. Methods Appl. Sci., 45 (2022), 12060-12076. https://doi.org/10.1002/mma.7495
29. 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.
30. 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
31. Kalman, D., Generalized Fibonacci numbers by matrix methods, Fibonacci Quart., 20(1) (1982), 73-76.
32. 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
33. Koshy, T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, Inc., New York, 2001.
34. Mamagani, A. B., Jafari, M., On properties of generalized quaternion algebra, Journal of Novel Applied Sciences, 2(12) (2013), 683-689.
35. Olariu, S., Complex Numbers in n-Dimensions, North-Holland Mathematics Studies, Amsterdam, 2002.
36. Pethe, S., Some identities for Tribonacci sequences, Fibonacci Quart., 26(2) (1988), 144-151.
37. 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
38. Pottmann, H.,Wallner, J., Computational Line Geometry, Springer-Verlag Berlin Heidelberg, New York, 2001.
39. Rochon, D., Shapiro, M., On algebraic properties of bicomplex and hyperbolic numbers, An Univ. Oradea Fasc. Mat., 11(71) (2004), 110.
40. 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
41. Shannon, A. G., Horadam, A. F., Some properties of third-order recurrence relations, Fibonacci Quart., 10(2) (1972), 135-146.
42. Sloane, N., The Online Encyclopedia of Integer Sequences, 1964, http://oeis.org/.
43. Sobczyk, G., The hyperbolic number plane, Coll. Math. J., 26(4) (1995), 268-280. https://doi.org/10.2307/2687027
44. Soykan, Y., On generalized third-order Pell numbers, Asian J. Adv. Res. Rep., 6(1) (2019), 1-18.
45. Soykan, Y., On generalized Grahaml numbers, J. Adv. Math. Comput. Sci., 35(2) (2020), 42-57. https://doi.org/10.9734/jamcs/2020/v35i230248
46. Soykan, Y., Generalized Pell-Padovan numbers, Asian J. Adv. Res. Rep., 11(2) (2020), 8-28. https://doi.org/10.9734/ajarr/2020/v11i230259
47. Soykan, Y., A note on binomial transform of the generalized 3-primes sequence, MathLAB Journal, 7(1) (2020), 168-190.
48. 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.
49. Soykan, Y., On generalized Narayana numbers, Int. J. Adv. Appl. Math. Mech., 7(3) (2020), 43-56.
50. 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
51. 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
52. Soykan, Y., Summing formulas for generalized Tribonacci numbers, Univers. J. Math. Appl., 3(1) (2020), 1-11. https://doi.org/10.32323/ujma.637876
53. Soykan, Y., A study on generalized (r, s, t)-numbers, MathLAB Journal, 7 (2020), 101-129.
54. Soykan, Y., On generalized Padovan numbers, Int. J. Adv. Appl. Math., 10(4) (2023), 72-90.
55. 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
56. 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
57. 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
58. Taşcı, D., Padovan and Pell-Padovan quaternions, Journal of Science and Arts, 42(1) (2018), 125-132.
59. 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
60. Waddill, M. E., Sacks, L., Another generalized Fibonacci sequence, Fibonacci Quart., 5(3) (1967), 209-222.
61. Yaglom, I. M., A Simple Non-Euclidean Geometry and Its Physical Basis, Springer-Verlag, New York, 1979.
62. Yalavigi, C. C., Properties of Tribonacci numbers, Fibonacci Quart., 10(3) (1972), 231-246.