The sequence of the hyperbolic \(k\)-Padovan quaternions

Downloads

DOI:

https://doi.org/10.26637/mjm1103/009

Abstract

This work introduces the hyperbolic \(k\)-Padovan quaternion sequence, performing the process of complexification of linear and recurrent sequences, more specifically of the generalized Padovan sequence. In this sense, there is the study of some properties around this sequence, deepening the investigative mathematical study of these numbers.

Keywords:

hyperbolic numbers, quaternions, \(k\)-Padovan sequence

Mathematics Subject Classification:

11B37, 11B39

Author Biographies

Renata Passos Machado Vieira, Post-Graduate Program in Education of the Nordeste Education Network – Polo RENOEN-UFC, Federal University of Ceara, Brazil.

Doctoral Student in Teaching (RENOEN-Pólo UFC). Master in Science and Mathematics Teaching from the Federal Institute of Education, Science and Technology of the State of Ceará. Teacher at the Secretary of Education of the State of Ceará.

Francisco Regis Vieira Alves, Federal Institute of Science and Technology Education of the State of Ceará

HEAD Professor at the Federal Institute of Science and Technology Education of the state of Ceará / IFCE - 40h/a with DE, of the Mathematics Degree course and CNPq Research Productivity Scholarship - Level (2020 - 2023). He has experience in the field of Mathematics, working mainly on the following subjects: Didactics of Mathematics, History of Mathematics, Real Analysis, Philosophy of Mathematics and Technologies applied to the teaching of Mathematics for higher education. With research focused on teaching Calculus I, II, III, Complex Analysis, ODE, Number Theory. And at the Open University of Brazil, with the distance teaching of Mathematics. He develops research aimed at teaching Multivariable Calculus and its internal transition. He also works in the Professional Master's in Science and Mathematics Teaching (ENCIMA) - UFC. Reviewer and ad hoc referee of the following journals: Vydya Educação, 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 Bulletin of Mathematics Education - BoEM and REMAT Magazine: Electronic Magazine of Mathematics. Editorial committee of the Boletim Cearense of Education and History of Mathematics (BOCEHM) and Coordinator of the Graduate Program in Science and Mathematics Teaching - PGECM/IFCE (academic). in the period 2015/2020 and Member of the Scientific Council of the journal ForSCience - IFMG. Reviewer of the EURASIA Journal of Mathematics, Science and Technology Education.

Paula Maria Machado Cruz Catarino, University of Trás-os-Montes and Alto Douro

PhD in Mathematics. Associate Professor of UTAD (Universidade de Trás-os-Montes e Alto Douro) with habilitation. Researcher of Research Centre CMAT-UTAD- Polo of CMAT of University of Minho and also Researcher of the Research Centre CIDTFF - Research Centre “Didactics and Technology in Education of Trainers. Currently Member of General Council of UTAD.

  • Pages: 324-331
  • Date Published: 01-07-2023
  • Vol. 11 No. 03 (2023): Malaya Journal of Matematik (MJM)

F.T. AYDIN, Hyperbolic k-Fibonacci Quaternions. https://arxiv.org/pdf/1812.00781.pdf, 2018.

P. CATARINO, On hyperbolic k-Pell quaternions sequences. Annales Mathematicae et Informaticae, 49 (2018), 61-73. DOI: https://doi.org/10.33039/ami.2018.05.005

P. CATARINO, Bicomplex k-Pell Quaternions. Computational Methods and Function Theory, 19(3)(2019), $65-76$. DOI: https://doi.org/10.1007/s40315-018-0251-5

F. CATONI, D. BOCCALETTI, R. CANNATA, V. CATONI AND P. ZAMPETTI, Hyperbolic Numbers in Geometry of MinkowskiSpace-Time, Springer, Heidelberg, p.3-23, (2011). DOI: https://doi.org/10.1007/978-3-642-17977-8_2

G. DATTOLI, S. LICCIARDI, R.M. PIDATELLA AND E. SABIA, Hybrid complex numbers: The matrix version, Adv. Appl. CliffordAlgebras, 28(3)(2018), 58. DOI: https://doi.org/10.1007/s00006-018-0870-y

S. HALICI AND A. KARATAS, On a generalization for fibonacci quaternions, Chaos Solitons, Fractals, 98(2017), 178-182. DOI: https://doi.org/10.1016/j.chaos.2017.03.037

S. HALICI AND A. KARATAS, Some Matrix Representations of Fibonacci Quaternions and Octonions. Advances in Applied Clifford Algebras, vol. 27, n. 2, p. 1233-1242, 2017. DOI: https://doi.org/10.1007/s00006-016-0661-2

S. HALICI, On Fibonacci Quaternions. Springer Science and Business Media LLC, p. 1-7, 2012. DOI: https://doi.org/10.1007/s00006-012-0337-5

A.F. HORADAM, Quaternion Recurrence relations, Ulam Quarterly, 2(2)(1993), $23-33$.

M.J. MENON, Sobre as origens das definições dos produtos escalar e vetorial, Revista Brasileira de Ensino de Física, 31(2)(2009), 1-11. DOI: https://doi.org/10.1590/S1806-11172009000200006

A.E. MOTTER AnD A.F. ROSA, Hyperbolic calculus, Adv. Appl. Clifford Algebras, 8(1)(1998), 109-128. DOI: https://doi.org/10.1007/BF03041929

Y. SOYKAN AND M. GOCEN, Properties of hyperbolic generalized Pell numbers, Notes on Number Theory and Discrete Mathematics, 26(4)(2020), 136-153. DOI: https://doi.org/10.7546/nntdm.2020.26.4.136-153

R.P.M. VIEIRA, Engenharia Didática (ED): o caso da Generalização e Complexificação da Sequência de Padovan ou Cordonnier. 266f. Dissertação de Mestrado Acadêmico em Ensino de Ciências e Matemática Instituto Federal de Educação, Ciência e Tecnologia do Estado do Ceará, 2020.

R.P.M. VIEIRA, F.R.V. ALVES AND P.M.C. CATARINO, A historic alanalys is of the padovan sequence, International Journal of Trends in Mathematics Education Research, 3(1)(2020), 8-12. DOI: https://doi.org/10.33122/ijtmer.v3i1.166

R.P.M. VIEIRA AND F.R.V. ALVES, Explorando a sequência de Padovan através de investigação histórica e abordagem epistemológica, Boletim GEPEM, 74(2019), 161-169. DOI: https://doi.org/10.4322/gepem.2019.012

R.P.M. VIEIRA AND F.R.V. ALVES, Os números duais de Padovan. Revista de Matemática da UFOP, $2(2019), 52-61$.

P. DAS, E. SAVAŞ AND S.K. GhOSAL, On generalizations of certain summability methods using ideals, Appl. Math. Lett., 24(2011), 1509-1514. DOI: https://doi.org/10.1016/j.aml.2011.03.036

H. FAST, Sur la convergence statistique, Colloq. Math., 2(1951), 241-244. DOI: https://doi.org/10.4064/cm-2-3-4-241-244

J.A. Fridy, On statistical convergence, Analysis, 5(1985), 301-313. DOI: https://doi.org/10.1524/anly.1985.5.4.301

J.A. FRIDY AND C. ORHAN, Lacunary statistical convergence, Pac J Math., 160(1)(1993), 43-51. DOI: https://doi.org/10.2140/pjm.1993.160.43

A.R. Freedman, J.J. Sember And M. Raphael, Some Cesàro-type summability spaces, Proc. Lond. Math. Soc., 37(1978), 508-520. DOI: https://doi.org/10.1112/plms/s3-37.3.508

J.A. Fridy AND C. ORhAn, Lacunary statistical summability, J. Math. Anal. Appl., 173(2)(1993), 497-504. DOI: https://doi.org/10.1006/jmaa.1993.1082

M. Gürdal and M.B Huban, On $mathcal{I}$-convergence of double sequences in the Topology induced by random 2-norms, Mat. Vesnik, 66(1)(2014), 73-83.

M. GÜRdal AND A. ŞAhiner, Extremal I -limit points of double sequences, Appl. Math. E-Notes, 8(2008), $131-137$.

Ö. KIşI, On invariant arithmetic statistically convergence and lacunary invariant arithmetic statistically convergence, Palest. J. Math., in press.

Ö. KışI, On I-lacunary arithmetic statistical convergence, J. Appl. Math. Informatics, in press.

P. KostYrko, M. Macaj And T. Šalát, II-convergence, Real Anal. Exchange, 26(2)(2000), 669-686. DOI: https://doi.org/10.2307/44154069

P. Kostyrko, M. Macaj, T. S̆alát and M. Sleziak, $mathcal{I}$-convergence and extremal $mathcal{I}$-limit points, Math. Slovaca, 55(2005), 443-464.

J. Li, Lacunary statistical convergence and inclusion properties between lacunary methods, Int. J. Math. Math. Sci., 23(3)(2000), 175-180. DOI: https://doi.org/10.1155/S0161171200001964

M. MURSAleen, Matrix transformation between some new sequence spaces, Houston J. Math., 9(1983), $505-509$.

M. MursaleEn, On finite matrices and invariant means, Indian J. Pure Appl. Math., 10(1979), 457-460.

A. Nabiev, S. PehlivAn AnD M. GÜRdal, On I-Cauchy sequences, Taiwanese J. Math., 11(2007), 569-566. DOI: https://doi.org/10.11650/twjm/1500404709

F. NURAy, Lacunary statistical convergence of sequences of fuzzy numbers, Fuzzy Sets and Systems, $mathbf{9 9 ( 3 ) ( 1 9 9 8 ) , 3 5 3 - 3 5 5 .}$

F. NURAY AND E. SAVAŞ, Invariant statistical convergence and $A$-invariant statistical convergence, Indian J. Pure Appl. Math., 25(3)(1994), 267-274.

F. NURAY AND E. SAVAŞ, On $sigma$ statistically convergence and lacunary $sigma$ statistically convergence, Math. Slovaca, 43(3)(1993), 309-315.

F. NURAY AND H. GöK AND U. Ulusu, $mathcal{I}_sigma$-convergence, Math. Commun., 16(2011), 531-538.

R.A. RAImI, Invariant means and invariant matrix methods of summability, Duke Math. J., 30(1963), 81-94. DOI: https://doi.org/10.1215/S0012-7094-63-03009-6

W.H. RuckLE, Arithmetical summability, J. Math. Anal. Appl., 396(2012), 741-748. DOI: https://doi.org/10.1016/j.jmaa.2012.06.048

T. ŠALÁt, On statistically convergent sequences of real numbers, Math. Slovaca, 30(1980), 139-150.

T. Šalát, B.C. Tripathy and M. Ziman, On some properties of $mathcal{I}$-convergence, Tatra Mt. Math. Publ., $mathbf{2 8}(2004), 279-286$.

E. SAVAŞ, Some sequence spaces involving invariant means, Indian J. Math., 31(1989), 1-8.

E. SAVAŞ, Strong $sigma$-convergent sequences, Bull. Calcutta Math. Soc., 81(1989), 295-300.

E. SAVAŞ AND R.F. PATTERSON, Lacunary statistical convergence of multiple sequences, Appl. Math. Lett., $mathbf{1 9 ( 6 ) ( 2 0 0 6 ) , 5 2 7 - 5 3 4 .}$

E. SAVAŞ AND M. GÜRDAL, $mathcal{I}$-statistical convergence in probabilistic normed space, Sci. Bull. Series A Appl. Math. Physics, 77(4)(2015), 195-204.

E. SAVAŞ AND M. GÜRDAL, Certain summability methods in intuitionistic fuzzy normed spaces, J. Intell. Fuzzy Syst., 27(4)(2014), 1621-1629. DOI: https://doi.org/10.3233/IFS-141128

E. SAVAŞ AND M. GÜRDAL, A generalized statistical convergence in intuitionistic fuzzy normed spaces, Science Asia, 41(2015), 289-294. DOI: https://doi.org/10.2306/scienceasia1513-1874.2015.41.289

P. SChAEfER, Infinite matrices and invariant means, Proc. Amer. Math. Soc., 36(1972), 104-110. DOI: https://doi.org/10.1090/S0002-9939-1972-0306763-0

B.C. Tripathy and B. HaZarika, I -monotonic and $mathcal{I}$-convergent sequences, Kyungpook Math. J., 51(2011), 233-239. DOI: https://doi.org/10.5666/KMJ.2011.51.2.233

U. ULUSU AND F. NURAY, Lacunary $mathcal{I}$-invariant convergence, Cumhuriyet Sci. J., 41(3)(2020), 617-624. DOI: https://doi.org/10.17776/csj.689877

T. YAYIng AND B. HAZARIKA, On arithmetical summability and multiplier sequences, Nat. Acad. Sci. Lett., 40(1)(2017), 43-46. DOI: https://doi.org/10.1007/s40009-016-0525-2

T. YaYing And B. HaZArika, On arithmetic continuity, Bol. Soc. Parana Mater., 35(1)(2017), 139-145. DOI: https://doi.org/10.5269/bspm.v35i1.27933

T. Yaying, B. Hazarika And H. ÇaKalli, New results in quasi cone metric spaces, J. Math. Comput. Sci., 16(2016), 435-444. DOI: https://doi.org/10.22436/jmcs.016.03.13

T. YaYing And B. HaZARIKA, On arithmetic continuity in metric spaces, Afr. Mat., 28(2017), 985-989. DOI: https://doi.org/10.1007/s13370-017-0498-4

T. Yaying and B. Hazarika, Lacunary Arithmetic Statistical Convergence, Nat. Acad. Sci. Lett., 43(6)(2020), 547-551. DOI: https://doi.org/10.1007/s40009-020-00910-6

  • National Council for Scientific and Technological Development - CNPq
  • Cearense Foundation for Support to Scientific and Technological Development (Funcap)
  • National Funds through FCT - Foundation for Science and Technology. I.P, within the scope of the UID / CED / 00194/2020 project

Metrics

Metrics Loading ...

Published

01-07-2023

How to Cite

Passos Machado Vieira, R., F. R. Vieira Alves, and P. M. Machado Cruz Catarino. “The Sequence of the Hyperbolic \(k\)-Padovan Quaternions”. Malaya Journal of Matematik, vol. 11, no. 03, July 2023, pp. 324-31, doi:10.26637/mjm1103/009.

Issue

Vol. 11 No. 03 (2023): Malaya Journal of Matematik (MJM)

Section

Articles

License

Copyright (c) 2023 Renata Passos Machado Vieira, Francisco Regis Vieira Alves, Paula Maria Machado Cruz Catarino

Creative Commons License

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

Crossref
Scopus
Google Scholar
Europe PMC