Perrin’s hyperbolic circular quaternions and Perrin’s hybridinomial numbers

Authors

DOI:

https://doi.org/10.14393/BEJOM-v3-n6-2022-65889

Keywords:

Números hiperbólicos duais, Hibrinomiais, Quatérnios hiperbólicos circulares, Sequência de Perrin

Abstract

Studies referring to linear and recurrent sequences are being carried out in the Pure Mathematics literature, thus expanding the universe of sequence evolution. With the study of Fibonacci’s hyperbolic circular quaternions, it was possible to define Perrin’s hy- perbolic circular quaternions for this research. In addition, Perrin’s hybrid numbers are introduced, based on Perrin’s hybrid and polynomial numbers.. Therefore, some algebraic properties of Perrin’s circular hyperbolic quaternions are studied, resulting in obtaining their respective generating function, Binet formula and matrix form.

Downloads

Download data is not yet available.

Author Biographies

Renata Passos Machado Vieira, Universidade Federal do Ceará

PhD student in Education at the Federal University of Ceará. Master's degree in Science and Mathematics Teaching from the Federal Institute of Education, Science, and Technology of the State of Ceará and teacher at the State Department of Education of Ceará.

Francisco Regis Vieira Alves, Instituto Federal de Educação Ciência e Tecnologia do Estado do Ceará

Professor TITULAR at the Federal Institute of Education, Science and Technology of the state of Ceará / IFCE - 40h/a with DE, in the Mathematics Teaching Degree course and Research Productivity Scholarship Holder from CNPq - Level (2020 - 2023). Has experience in the area of Mathematics, mainly focusing on the following topics: Mathematics Didactics, History of Mathematics, Real Analysis, Philosophy of Mathematics, and Technologies applied to mathematics teaching at the higher education level. Conducts research focused on the teaching of Calculus I, II, III, Complex Analysis, Ordinary Differential Equations, and Number Theory. Also involved in the Open University of Brazil, teaching Mathematics through distance education. Conducts research directed towards teaching Multivariable Calculus and its internal transition. Also participates in the Professional Master's Program in Teaching Sciences and Mathematics (ENCIMA) - UFC. Serves as a reviewer and ad hoc reviewer for the following journals: Vydya Education, Sinergia - IFSP, Rencima - Journal of Science and Mathematics Teaching, Journal of the Geogebra Institute of São Paulo, Tear - Journal of Education, Science, and Technology, Online Journal of Mathematics Education – Bo EM, and REMAT: Electronic Journal of Mathematics. Editorial Committee member of the Cearense Bulletin of Education and History of Mathematics (BOCEHM) and Coordinator of the Graduate Program in Teaching Sciences and Mathematics - PGECM/IFCE (academic) from 2015 to 2020. Also, a member of the Scientific Council of the journal For Science - IFMG. Evaluator for the EURASIA Journal of Mathematics, Science, and Technology Education.

 

Paula Maria Machado Cruz Catarino, Universidade de Trás-os-Montes e Alto Douro

PhD em Matemática. Professora Associada da UTAD (Universidade de Trás-os-Montes e Alto Douro) com habilitação. Investigadora do Centro de Investigação CMAT-UTAD- Polo do CMAT da Universidade do Minho e também Investigadora do Centro de Investigação CIDTFF - Centro de Investigação “Didática e Tecnologia na Formação de Formadores. Atualmente Membro do Conselho Geral da UTAD.

References

ALVES, F. R. V.; CATARINO, P. M. M. C.; VIEIRA, R. P. M.; MANGUEIRA, M. dos S. Teaching Recurrent Sequences in Brazil using Historical facts and graphicalillustrations,Acta Didactica Napocensia, vol. 13, n. 1, p. 87-104, 2020.

ALVES, F. R. V.; VIEIRA, R. P. M.; CATARINO, P. M. M. C. Visualizing the Newtons Fractal from the Recurring Linear Sequence with Google Colab: An Exampleof Brazil X Portugal Research, International Electronic Journal of Mathematics Education, vol. 15, n. 3, p. 1-19, 2020.

AYDIN, F. T. Circular-hyperbolic Fibonacci quaternions. Notes on Number Theory and Discrete Mathematics, vol. 26, n. 2, p. 167-176, 2020.

PENNESTRI, E.; STEFANELLI, R. Linear Algebra and Numerical Algorithms Using Dual Numbers. Multibody System Dynamics, vol. 18, n. 3, p. 323-344,2007.

CIHAN, A.; AZAK, A. Z.; GUNGOR, M. A.; TOSUN, M.V. A study of Dual Hyperbolic Fibonacci and Lucas numbers.An. St. Univ. Ovidius Constanta, vol. 27,n. 1, p. 35-48, 2019.

DATTOLI, G.; LICCIARDI, S.; PIDATELLA, R. M.; SABIA, E. Hybrid complex numbers: The matrix version. Adv. Appl. Clifford Algebras, vol. 28, n. 3, p. 58,2018.

HORZUM, T; KOCER, E. G. On some properties of Horadam polynomials. Inter-national Mathematical Forum, vol. 4, n.25, p. 1243-1252, 2009.

HAUKKANEN, P. A Note On Horadam?s Sequence.The Fibonacci Quart., vol.40, n. 4, p. 358-361, 2002.

JANCEWICZ, B. The extended Grassmann algebra of R3, in Clifford (Geometric) Algebras with Applications and Engineering. Birkhauser, Boston, p. 389-421, 1996.

KIZILATES, C. A note on Horadam hybrinomials. 2020010116 (doi:10.20944/preprints202001.0116.v1), p. 1-10, 2020.

KNUTH, D. E. The Art of Computer Programming. Addison-Wesley, Reading, MA, vol.3, 2nd edition, 417, 1998.

MANGUEIRA, M. et al. The hybrid numbers of Padovan and some identities.2020010116 . Annales Mathematicae Silesianae, vol. 34, n. 2, p. 256-267, 2020.

ORHAN, D.; HAMZA, M. On the Split (s,t)-Padovan and (s,t)-Perrin Quaternions. International Journal of Applied Mathematics and Informatics, vol. 13, p. 25-28, 2019.

OLIVEIRA, R. R. de O. Engenharia Didática sobre o Modelo de Comple-xificação da Sequência Generalizada de Fibonacci: Relações Recorrentes N-dimensionais e Representações Polinomiais e Matriciais. Mestrado Acadêmico em Ensino de Ciências e Matemática - Instituto Federal de Educação, Ciência e Tecnologia do Estado do Ceará (IFCE), 2018.

OZDEMIR, M.; HAMZA, M. Introduction to Hybrid Numbers. Adv. Appl. Clifford Algebras, vol. 28, n. 11, 2018.

SHOHAN, M. On grassmann’s products and clifford’s dual unit. International Symposium on History of Machines and Mechanisms Proceedings HMM, p.173-180, 2000.

SOBCZYK, G. The Hyperbolic Number Plane. The College Mathematics Journal,vol. 26, n. 4, p. 268-280, 1995.

SOYKAN, Y.; GOCEN, M. Properties of hyperbolic generalized Pell numbers. Notes on Number Theory and Discrete Mathematics, vol. 26, n. 4, p. 136-153, 2020.

SUGUMARAN, A.; RAJESH, K. Os números duais de Padovan. I International Journal of Pure and Applied Mathematics, vol. 114, n. 6, p. 131-137, 2017.

VIEIRA, R. P. M.; ALVES, F. R. V; CATARINO, P. M. C. A historic alanalys is of the padovan sequence. International Journal of Trends in Mathematics Education Research, vol. 3, n. 1, p. 8-12, 2020.

VIEIRA, R. P. M.; ALVES, F. R. V. Explorando a sequência de Padovan através de investigação histórica e abordagem epistemológica. Boletim GEPEM, vol. 74, p.161-169, 2019.

Downloads

Published

2022-12-23

How to Cite

VIEIRA, R. P. M.; VIEIRA ALVES, F. R.; CATARINO, P. M. M. C. Perrin’s hyperbolic circular quaternions and Perrin’s hybridinomial numbers. BRAZILIAN ELECTRONIC JOURNAL OF MATHEMATICS, Uberlândia, v. 3, n. 6, 2022. DOI: 10.14393/BEJOM-v3-n6-2022-65889. Disponível em: https://seer.ufu.br/index.php/BEJOM/article/view/65889. Acesso em: 15 jan. 2025.

Issue

Vol. 3 No. 6 (2022): Brazilian Electronic Journal of Mathematics

Section

Articles - Pure Mathematics

License

Copyright (c) 2022 BRAZILIAN ELECTRONIC JOURNAL OF MATHEMATICS

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

The published articles are licensed under the CreativeCommons CCBY-NC/4.0 version. By submitting the material for publication, the authors automatically waive their copyright, agree to the editorial guidelines of the journal, and assume that the text has been properly reviewed. Simultaneous submission of articles to other journals is prohibited, as is the translation of articles published in the journal into another language without proper authorization.

Most read articles by the same author(s)