Some New Identities with Respect to Bihyperbolic Fibonacci and Lucas Numbers
AbstractIn this paper, we define bihyperbolic Lucas numbers, bihyperbolic generalized Fibonacci numbers and give some algebraic properties of these numbers. Then, some identities concerning conjugations, Honsberger's identity, negabihyperbolic numbers for bihyperbolic Fibonacci number and bihyperbolic Lucas numbers have been derived. Finally, well-known identities have been represented such as Cassini's, Catalan's, d'Ocagne's identities and Binet formula for bihyperbolic Lucas and bihyperbolic generalized Fibonacci numbers. Also, special cases of these identities and formulas have been given.
. W.R. Hamilton. Lectures on Quaternions. Dublin: Hodges and Smith, 1853.
. J. Cockle. “On certain functions resembling quaternions and on a new imaginary in algebra.” The London, Edinburg and Dublin Philosophical Magazine and Journal of Science, vol. 33, no. 3, pp. 435-439, 1848.
. C. Segre. “Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici.” Mathematische Annalen, vol 40, pp. 413–467, 1892.
. S. Olariu, Complex Numbers in n Dimensions. North-Holland Mathematics Studies, vol. 190, Amsterdam, Boston: Elsevier, 2002, pp. 51-148.
. 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. Basel, Boston, Berlin: Birkhäuser Verlag, 2008.
. D. Alpay, M.E. Luna-Elizarraras, M. Shapiro, D.C. Struppa. Basics of Functional Analysis with Bicomplex Scalars and Bicomplex Schur Analysis. Springer Briefs in Mathematics. Springer, 2014.
. F. Colombo, I. Sabadin, D.C. Struppa. “Bicomplex holomorphic functional calculus.” Mathematische. Nachrichten, vol. 287, no. 13, pp. 1093-1105, 2013.
. R. Gervais Lavoie, L. Marchildon, D. Rochon. “Infinite-dimensional bicomplex Hilbert spaces.” Annals of Functional Analysis, vol. 1, no. 2, pp. 75-91, 2010.
. R. Gervais Lavoie, L. Marchildon, D. Rochon. “Finite-dimensional bicomplex Hilbert spaces.” Advances in Applied Cliﬀord Algebras, vol. 21, no. 3, pp. 561-581, 2011.
. R. Kumar, H. Saini. “Topological bicomplex modules.” Advances in Applied Cliﬀord Algebras, vol. 26, no. 4, pp. 1249-1270, 2016.
. R. Kumar, K. Singh, H. Saini, S. Kumar. “Bicomplex weighted Hardy spaces and bicomplex -algebras.” Advances in Applied Cliﬀord Algebras, vol. 26, no. 1, pp. 217-235, 2016.
. M.E. Luna-Elizarraras, M. Shapiro, D.C. Struppa, A. Vajiac. “Bicomplex numbers and their elementary functions.” Cubo, vol. 14, no. 2, pp. 61-80, 2012.
. M.E. Luna-Elizarraras, C.O. Perez-Regalado, M. Shapiro. “On linear functionals and Hahn-Banach theorems for hyperbolic and bicomplex modules.” Advances in Applied Cliﬀord Algebras, vol. 24, pp. 1105-1129, 2014.
. M.E. Luna-Elizarraras, M. Shapiro, D.C. Struppa, A. Vajiac. Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers. Frontiers in Mathematics, New York: Springer, 2015.
. M.E. Luna-Elizarraras, C.O. Perez-Regalado, M. Shapiro. “On the bicomplex Gleason-Kahane Zelazko theorem.” Complex Analysis and Operator Theory, vol. 10, no. 2, pp. 327-352, 2016.
. K. Manzoor. “A note on bicomplex linear operators on bicomplex Hilbert spaces.” International Journal of Mathematics Trends and Technology, vol. 36, no. 3, pp. 218-224, 2016.
. D. Rochon, M. Shapiro. “On algebraic properties of bicomplex and hyperbolic numbers.” Annals of Oradea University-Mathematics Fascicola, vol. 11, pp. 71-110, 2004.
. D. Rochon, S. Tremblay. “Bicomplex quantum mechanics II: the Hilbert space. ” Advances in Applied Cliﬀord Algebras, vol. 16, no. 2, pp. 135-157, 2006.
. S.K. Nurkan, İ.A. Güven. “A note on bicomplex Fibonacci and Lucas numbers.” International Journal of Pure Applied Mathematics, vol. 120, no. 3, pp. 365-377, 2018.
. F.T. Aydın. “Bicomplex Fibonacci quaternions.” Chaos Solitons and Fractals, vol. 106, pp. 147-153, 2018.
. S. Halıcı, Ş. Çürük. “On bicomplex numbers with coefﬁcients from the complex Fibonacci sequence.” Notes on Number Theory and Discrete Mathematics, vol. 25, no. 3, pp. 126–137, 2019.
. S. Halıcı. On Bicomplex Fibonacci Numbers and Their Generalization. Models and Theories in Social Systems, Studies in Systems, Decision and Control, Vol. 179, pp.509-524, Springer Cham, 2019.
. A.A. Pogorui, R.M. Rodrigez-Dagnino, R.D. Rodrigez-Said. “On the set of zeros of bihyperbolic polynomials.” Complex Variables and Elliptic Equations, vol. 53, no. 7, pp. 685-690, 2008.
. M. Bilgin, S. Ersoy. “Algebraic properties of bihyperbolic numbers.” Advances in Applied Cliﬀord Algebras, vol. 30, no. 13, pp. 1–17, 2020.
. N. Gürses, N., G.Y. Şentürk, S. Yüce. “A Study on Dual-Generalized Complex and Hyperbolic -Generalized Complex Numbers.” Gazi University Journal of Science, vol. 34, no. 1, pp. 180-194, 2021.
. D. Brod, A. Syznal-Liana, I. Wloch. “On Some Combinatorial Properties of Bihyperbolic Numbers of the Fibonacci Type.” Mathematical Methods in the Applied Sciences, vol. 44, no. 6, pp. 4607-4615, 2021.
. A.F. Horadam. “A generalized Fibonacci sequence.” The American Mathematical Monthly, vol. 68, pp. 455-459, 1961.
. S. Vajda. Fibonacci and Lucas Numbers and The Golden Section. England: Ellis Horwood Limited Publ., 1989.
. R.A.Dunlap. The Golden Ratio and Fibonacci Numbers. NewJersey, London, Singapore, HongKong: World Scientific Pub. Co. Pte. Ltd., 1997.
. E.W. Weisstein. “Fibonacci Number”, MathWorld, Internet:http://mathworld.wolfram.com/Fibonacci Number.html.
. D. Knuth. “Negafibonacci Numbers and Hyperbolic Plane”. The Annual Meeting of the Math. Association of America, Fairmonth Hotel, San Jose, 2008.
. T. Koshy. Fibonacci and Lucas Numbers with Applications. USA: A Wiley-Intersience Publication, 2001.
. F. Köken, D. Bozkurt. “On Lucas numbers by the matrix method.” Hacettepe Journal of Mathematics and Statistics vol. 39, no. 4, pp. 471–475, 2010.
Copyright (c) 2021 International Journal of Sciences: Basic and Applied Research (IJSBAR)
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors who submit papers with this journal agree to the following terms.