Mathematica Bohemica, Vol. 149, No. 1, pp. 75-85, 2024
On generalized bihyperbolic Mersenne numbers
Dorota Bród, Anetta Szynal-Liana
Received June 18, 2022. Published online March 3, 2023.
Abstract: In this paper, a new generalization of Mersenne bihyperbolic numbers is introduced. Some of the properties of presented numbers are given. A general bilinear index-reduction formula for the generalized bihyperbolic Mersenne numbers is obtained. This result implies the Catalan, Cassini, Vajda, d'Ocagne and Halton identities. Moreover, generating function and matrix generators for these numbers are presented.
References: [1] M. Bilgin, S. Ersoy: Algebraic properties of bihyperbolic numbers. Adv. Appl. Clifford Algebr. 30 (2020), Article ID 13, 17 pages. DOI 10.1007/s00006-019-1036-2 | MR 4054825 | Zbl 1442.30049
[2] P. Catarino, H. Campos, P. Vasco: On the Mersenne sequence. Ann. Math. Inform. 46 (2016), 37-53. MR 3607003 | Zbl 1374.11020
[3] F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti: The Mathematics of Minkowski Space-Time: With an Introduction to Commutative Hypercomplex Numbers. Frontiers in Mathematics. Birkhäuser, Basel (2008). DOI 10.1007/978-3-7643-8614-6 | MR 2411620 | Zbl 1151.53001
[4] M. Chelgham, A. Boussayoud: On the $k$-Mersenne-Lucas numbers. Notes Number Theory Discrete Math. 27 (2021), 7-13. DOI 10.7546/nntdm.2021.27.1.7-13
[5] J. Cockle: On certain functions resembling quaternions, and on a new imaginary in algebra. Phil. Mag. (3) 33 (1848), 435-439. DOI 10.1080/14786444808646139
[6] J. Cockle: On a new imaginary in algebra. Phil. Mag. (3) 34 (1849), 37-47. DOI 10.1080/14786444908646169
[7] J. Cockle: On the symbols of algebra, and on the theory of tessarines. Phil. Mag. (3) 34 (1849), 406-410. DOI 10.1080/14786444908646257
[8] J. Cockle: On impossible equations, on impossible quantities, and on tessarines. Phil. Mag. (3) 37 (1850), 281-283. DOI 10.1080/14786445008646598
[9] A. Daşdemir, G. Bilgici: Gaussian Mersenne numbers and generalized Mersenne quaternions. Notes Number Theory Discrete Math. 25 (2019), 87-96. DOI 10.7546/nntdm.2019.25.3.87-96
[10] P. Ochalik, A. Włoch: On generalized Mersenne numbers, their interpretations and matrix generators. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 72 (2018), 69-76. DOI 10.17951/a.2018.72.1.69-76 | MR 3832418 | Zbl 1441.11027
[11] S. Olariu: Commutative complex numbers in four dimensions. Complex Numbers in $n$ Dimensions. North-Holland Mathematics Studies 190. Elsevier, Amsterdam (2002), 51-147. DOI 10.1016/S0304-0208(02)80004-4 | MR 1922267 | Zbl 1023.30001
[12] A. A. Pogorui, R. M. Rodríguez-Dagnino, R. D. Rodríguez-Said: On the set of zeros of bihyperbolic polynomials. Complex Var. Elliptic Equ. 53 (2008), 685-690. DOI 10.1080/17476930801973014 | MR 2431350 | Zbl 1158.30300
[13] D. Rochon, M. Shapiro: On algebraic properties of bicomplex and hyperbolic numbers. An. Univ. Oradea, Fasc. Mat. 11 (2004), 71-110. MR 2127591 | Zbl 1114.11033
[14] A. M. Sergeev: Generalized Mersenne matrices and Balonin's conjecture. Autom. Control Comput. Sci. 48 (2014), 214-220. DOI 10.3103/S0146411614040063
[15] Y. Soykan: A study of generalized Mersenne numbers. J. Progress. Research Math. 18 (2021), 90-108.
Affiliations: Dorota Bród (corresponding author), Anetta Szynal-Liana, Rzeszow University of Technology, The Faculty of Mathematics and Applied Physics, al. Powstańców Warszawy 12, 35-959 Rzeszów, Rzeszów, Poland, e-mail: dorotab@prz.edu.pl, aszynal@prz.edu.pl
