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On The Properties Of The Complex Fibonacci And Lucas Numbers With Rational Subscript Via Roots Of The Fibonacci Matrix

Year 2019, , 148 - 157, 24.03.2019
https://doi.org/10.18185/erzifbed.431628

Abstract

In this study, we exploit general techniques from matrix theory to establish some identities for the complex Fibonacci and Lucas numbers with rational subscripts of the forms  and . For this purpose, we establish matrix functions  and  of the Fibonacci matrix  of order  for integer odd nand discuss some relations between two special matrices functions  and , respectively. Also, some identities related to the complex Fibonacci and Lucas numbers with rational subscripts of the forms  and  are given for every integer odd and, respectively.

References

  • Arslan, S., Köken, F. (2017). “Some Identities For The Fibonacci And Lucas Sequences With Rational Subscript Via Matrix Methods”, Bulletin Of Mathematics And Statistics Research, 5 (4), 81-87.
  • Bicknell M. and Hoggatt, Jr. V. E. (1963). “Fibonacci Matrices and Lambda Functions”, Fibonacci Quarterly, 1 (2), 47-52.
  • Bicknell, M. (1965). “Fibonacci fantasy: The square root of the Q matrix”, Fibonacci Quarterly, 3 (1), 67-71.
  • Gantmacher, F.R. (1960). “The Theory of Matrices”, Volume 1. Chelsea, New York.
  • Gould, H.W. 1981. “A history of the Fibonacci Q-matrix and a higher-dimensional problem”, Fibonacci Quarterly, 19 (3), 250-257.
  • Halsey, E. (1965). “The Fibonacci Number where u is not an Integer”, Fibonacci Quarterly, 3 (2), 147-152.
  • Halici S., Akyüz Z. (2016). “Fibonacci and Lucas Sequences at Negative Indices”, Konuralp Journal of Mathematics, 4 (1), 172-178.
  • Higham, N.J. (2008). “Functions of Matrices : Theory and Computation”, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.
  • Horadam, A. F. and Shannon, A.G. (1988). “Fibonacci and Lucas Curves”, Fibonacci Quarterly, 26 (1), 3-13.
  • Hoggatt, Jr. V. E. and Bicknell, M. (1964). Some new Fibonacci Identities, Fibonacci Quarterly, 2, (1), 29-31.
  • Koshy, T. (2001). “Fibonacci and Lucas Numbers with Applications”, John Wiley & Sons Inc.
  • Parker, F.D. (1968). “A Fibonacci Function”, Fibonacci Quarterly, 6 (1), 1-2.
  • Richard, A.J. (1991). “Generalized Complex Fibonacci and Lucas Functions”, Fibonacci Quarterly, 29 (1), 13-18.
  • Vajda, S. (1989). “Fibonacci, Lucas numbers, and the golden section Theory and Applications”. Dover Publications Inc, Mineola N.Y., ISBN-13: 978-0486462769.
  • Wituła, R. (2011). “Fibonacci and Lucas Numbers for Real Indices and Some Applications”, Acta Physica Polonica A, 120 (4), 755-758.

Fibonacci Matrisinin Kökleri Aracılığıyla Fibonacci Ve Lucas Sayılarının Özellikleri Üzerine

Year 2019, , 148 - 157, 24.03.2019
https://doi.org/10.18185/erzifbed.431628

Abstract

Bu çalışmada,  ve  formlarındaki rasyonel indisli kompleks Fibonacci ve Lucas sayıları için bazı eşitlikler oluşturmak için matris teorisinden genel tekniklerden faydalanırız. Bu amaçla, tek ntam sayıları için  mertebeden Fibonacci  matrisinin and  matris fonksiyonlarını kurar ve sırasıyla  and  iki özel matris fonksiyonu arasındaki bazı ilişkileri ele alırız. Sırasıyla, tek tam sayıları ve  için  ve  formlarındaki rasyonel indisli kompleks Fibonacci ve Lucas sayıları ile ilgili bazı eşitlikler veririz.

References

  • Arslan, S., Köken, F. (2017). “Some Identities For The Fibonacci And Lucas Sequences With Rational Subscript Via Matrix Methods”, Bulletin Of Mathematics And Statistics Research, 5 (4), 81-87.
  • Bicknell M. and Hoggatt, Jr. V. E. (1963). “Fibonacci Matrices and Lambda Functions”, Fibonacci Quarterly, 1 (2), 47-52.
  • Bicknell, M. (1965). “Fibonacci fantasy: The square root of the Q matrix”, Fibonacci Quarterly, 3 (1), 67-71.
  • Gantmacher, F.R. (1960). “The Theory of Matrices”, Volume 1. Chelsea, New York.
  • Gould, H.W. 1981. “A history of the Fibonacci Q-matrix and a higher-dimensional problem”, Fibonacci Quarterly, 19 (3), 250-257.
  • Halsey, E. (1965). “The Fibonacci Number where u is not an Integer”, Fibonacci Quarterly, 3 (2), 147-152.
  • Halici S., Akyüz Z. (2016). “Fibonacci and Lucas Sequences at Negative Indices”, Konuralp Journal of Mathematics, 4 (1), 172-178.
  • Higham, N.J. (2008). “Functions of Matrices : Theory and Computation”, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.
  • Horadam, A. F. and Shannon, A.G. (1988). “Fibonacci and Lucas Curves”, Fibonacci Quarterly, 26 (1), 3-13.
  • Hoggatt, Jr. V. E. and Bicknell, M. (1964). Some new Fibonacci Identities, Fibonacci Quarterly, 2, (1), 29-31.
  • Koshy, T. (2001). “Fibonacci and Lucas Numbers with Applications”, John Wiley & Sons Inc.
  • Parker, F.D. (1968). “A Fibonacci Function”, Fibonacci Quarterly, 6 (1), 1-2.
  • Richard, A.J. (1991). “Generalized Complex Fibonacci and Lucas Functions”, Fibonacci Quarterly, 29 (1), 13-18.
  • Vajda, S. (1989). “Fibonacci, Lucas numbers, and the golden section Theory and Applications”. Dover Publications Inc, Mineola N.Y., ISBN-13: 978-0486462769.
  • Wituła, R. (2011). “Fibonacci and Lucas Numbers for Real Indices and Some Applications”, Acta Physica Polonica A, 120 (4), 755-758.
There are 15 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Makaleler
Authors

Fikri Köken

Publication Date March 24, 2019
Published in Issue Year 2019

Cite

APA Köken, F. (2019). On The Properties Of The Complex Fibonacci And Lucas Numbers With Rational Subscript Via Roots Of The Fibonacci Matrix. Erzincan University Journal of Science and Technology, 12(1), 148-157. https://doi.org/10.18185/erzifbed.431628