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Investigation of Matrices Q^(n._L ), M^(n._L ) and R^(n._L ) and Some Related Identities

Year 2022, Volume: 26 Issue: 4, 677 - 686, 31.08.2022

İbrahim Gökcan Ali Hikmet Değer

https://doi.org/10.16984/saufenbilder.1034057

Abstract

In this study, it is aimed to use the Lorenz matrix multiplication to find the n^th powers of some special matrices and to reach the quadratic equations and characteristic roots of the matrices obtained in this way. In addition, it is aimed to contribute literature to the studies in the field by reaching some identities.

Keywords

Characteristic Roots Fibonacci and Lucas Numbers Lorentz Matrix Multiplication Identity Quadratic Equations

References

  • [1] B.U. Alfred, “An Introduction to Fibonacci Discovery”,The Fibonacci Association, California, 1965.
  • [2] M. Bicknell, V.E. Hoggatt, “A Primer for the Fibonacci Numbers”, The Fibonacci Quarterly, 1973.
  • [3] I.D. Ruggles,“Some Fibonacci results using Fibonacci-type sequences”,The Fibonacci Quarterly, vol.1, pp.75-80, 1963.
  • [4] D. Tasci, E. Kilic,“On the order-k generalized Lucas numbers”,Applied Mathematics and Computation, vol.155, pp.637-641,2004.
  • [5] E. Kilic, D. Tasci, “On the generalized order-k Fibonacci and Lucas numbers”, Rocky Mountain Journal of Mathematics, vol.36, pp.1915-1926, 2006.
  • [6] C. Kızılateş, N. Tuglu, “A New generalization of convolved (p,q)-Fibonacci and (p,q)-Lucas polinomials”, Journal of Mathematical and Computational Science, vol.7, pp.995-1005, 2017.
  • [7] F. Qi, C. Kızılateş, W.S. Du, “A closed formula for the Horadam polynomials in terms of a tridiagonal determinant”, Symettery, vol. 11, pp. 782, 2019.
  • [8] C. Kızılateş, “New families of Horadam numbers associated with finite operators and their applications”, Mathematical Methods in the Applied Sciences, vol. 44, pp. 14371-14381, 2021.
  • [9] C. Kızılateş, W.S. Du, F. Qi “Several determinantal expressions of generalized tribonacci polynomials and sequences, Tamkang Journal of Mathematics, vol. 53, pp. 277-291, 2022.
  • [10] H.W. Gould, “A history of the Fibonacci Q-matrix and a higher-dimensional problem”, Fibonacci Quarterly, vol.9, pp.7-250, 1981.
  • [11] S. L. Basin, V. E. Hoggatt, “A primer on the Fibonacci sequence”,Part II, Fibonacci Quarterly, vol.2, pp.61-68, 1963.
  • [12] C.H. King, “Some properties of the Fibonacci numbers”, Master Thesis, San Jose State College, 1960.
  • [13] J.L. Brenner,“Lucas’ matrix”,The American Mathematical Monthly, vol.58, pp.220-221, 1951.
  • [14] J.S. Frame, “Continued fractions and matrices”, The American Mathematical Monthly,vol.56,pp.98-103,1949.
  • [15] H. Schwerdtfeger, “Geometry of complex numbers, mathematical expositions”, University of Totonto Press,1962.
  • [16] E.Jacobsthal, “Fibonaccische polynome und kreistheilungsgleichungen”,Sitzungsberischte der Berliner Math. Gesellschaft, vol.17 pp. 43-47, 1919-20.
  • [17] G.K. White, “On generators and defining relations for the unimodular group M2”,The American Mathematical Monthly, vol.71, pp.743-748, 1964.
  • [18] M.Bicknell, “Fibonacci fantasy:The square root of the Q matrix”,Fibonacci Quarterly, vol.3,pp.67-71, 1965.
  • [19] T.Koshy, “Fibonacci and Lucas numbers with applications”, Wiley-Interscience Publication, New York, 2001.
  • [20] H. Gündoğan, O. Keçilioğlu, “Lorentzian matrix multiplication and the motions on Lorentzian plane”,Glasnık Matematic ki, vol.41, pp.329-334, 2006.
  • [21] J.G. Ratcliffe, “Foundations of Hyperbolic manifolds”,Springer-Verlag, New York, 1994.
  • [22] O. Keçilioğlu, H. Gündoğan, “Pseudo matrix multiplication”,Communications Faculty of Science University of Ankara Series A1 Mathematics and Statistics, vol.66, pp.37-43, 2017.
  • [23] J.H. Halton, “On a general Fibonacci identity”,The Fibonacci Quarterly, vol.3, pp.31-43, 1965.

Year 2022, Volume: 26 Issue: 4, 677 - 686, 31.08.2022

İbrahim Gökcan Ali Hikmet Değer

https://doi.org/10.16984/saufenbilder.1034057

Abstract

References

  • [1] B.U. Alfred, “An Introduction to Fibonacci Discovery”,The Fibonacci Association, California, 1965.
  • [2] M. Bicknell, V.E. Hoggatt, “A Primer for the Fibonacci Numbers”, The Fibonacci Quarterly, 1973.
  • [3] I.D. Ruggles,“Some Fibonacci results using Fibonacci-type sequences”,The Fibonacci Quarterly, vol.1, pp.75-80, 1963.
  • [4] D. Tasci, E. Kilic,“On the order-k generalized Lucas numbers”,Applied Mathematics and Computation, vol.155, pp.637-641,2004.
  • [5] E. Kilic, D. Tasci, “On the generalized order-k Fibonacci and Lucas numbers”, Rocky Mountain Journal of Mathematics, vol.36, pp.1915-1926, 2006.
  • [6] C. Kızılateş, N. Tuglu, “A New generalization of convolved (p,q)-Fibonacci and (p,q)-Lucas polinomials”, Journal of Mathematical and Computational Science, vol.7, pp.995-1005, 2017.
  • [7] F. Qi, C. Kızılateş, W.S. Du, “A closed formula for the Horadam polynomials in terms of a tridiagonal determinant”, Symettery, vol. 11, pp. 782, 2019.
  • [8] C. Kızılateş, “New families of Horadam numbers associated with finite operators and their applications”, Mathematical Methods in the Applied Sciences, vol. 44, pp. 14371-14381, 2021.
  • [9] C. Kızılateş, W.S. Du, F. Qi “Several determinantal expressions of generalized tribonacci polynomials and sequences, Tamkang Journal of Mathematics, vol. 53, pp. 277-291, 2022.
  • [10] H.W. Gould, “A history of the Fibonacci Q-matrix and a higher-dimensional problem”, Fibonacci Quarterly, vol.9, pp.7-250, 1981.
  • [11] S. L. Basin, V. E. Hoggatt, “A primer on the Fibonacci sequence”,Part II, Fibonacci Quarterly, vol.2, pp.61-68, 1963.
  • [12] C.H. King, “Some properties of the Fibonacci numbers”, Master Thesis, San Jose State College, 1960.
  • [13] J.L. Brenner,“Lucas’ matrix”,The American Mathematical Monthly, vol.58, pp.220-221, 1951.
  • [14] J.S. Frame, “Continued fractions and matrices”, The American Mathematical Monthly,vol.56,pp.98-103,1949.
  • [15] H. Schwerdtfeger, “Geometry of complex numbers, mathematical expositions”, University of Totonto Press,1962.
  • [16] E.Jacobsthal, “Fibonaccische polynome und kreistheilungsgleichungen”,Sitzungsberischte der Berliner Math. Gesellschaft, vol.17 pp. 43-47, 1919-20.
  • [17] G.K. White, “On generators and defining relations for the unimodular group M2”,The American Mathematical Monthly, vol.71, pp.743-748, 1964.
  • [18] M.Bicknell, “Fibonacci fantasy:The square root of the Q matrix”,Fibonacci Quarterly, vol.3,pp.67-71, 1965.
  • [19] T.Koshy, “Fibonacci and Lucas numbers with applications”, Wiley-Interscience Publication, New York, 2001.
  • [20] H. Gündoğan, O. Keçilioğlu, “Lorentzian matrix multiplication and the motions on Lorentzian plane”,Glasnık Matematic ki, vol.41, pp.329-334, 2006.
  • [21] J.G. Ratcliffe, “Foundations of Hyperbolic manifolds”,Springer-Verlag, New York, 1994.
  • [22] O. Keçilioğlu, H. Gündoğan, “Pseudo matrix multiplication”,Communications Faculty of Science University of Ankara Series A1 Mathematics and Statistics, vol.66, pp.37-43, 2017.
  • [23] J.H. Halton, “On a general Fibonacci identity”,The Fibonacci Quarterly, vol.3, pp.31-43, 1965.
There are 23 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

İbrahim Gökcan 0000-0002-6933-8494

Ali Hikmet Değer 0000-0003-0764-715X

Publication Date August 31, 2022
Submission Date December 8, 2021
Acceptance Date May 30, 2022
Published in Issue Year 2022 Volume: 26 Issue: 4

Cite

APA Gökcan, İ., & Değer, A. H. (2022). Investigation of Matrices Q^(n._L ), M^(n._L ) and R^(n._L ) and Some Related Identities. Sakarya University Journal of Science, 26(4), 677-686. https://doi.org/10.16984/saufenbilder.1034057
AMA Gökcan İ, Değer AH. Investigation of Matrices Q^(n._L ), M^(n._L ) and R^(n._L ) and Some Related Identities. SAUJS. August 2022;26(4):677-686. doi:10.16984/saufenbilder.1034057
Chicago Gökcan, İbrahim, and Ali Hikmet Değer. “Investigation of Matrices Q^(n._L ), M^(n._L ) and R^(n._L ) and Some Related Identities”. Sakarya University Journal of Science 26, no. 4 (August 2022): 677-86. https://doi.org/10.16984/saufenbilder.1034057.
EndNote Gökcan İ, Değer AH (August 1, 2022) Investigation of Matrices Q^(n._L ), M^(n._L ) and R^(n._L ) and Some Related Identities. Sakarya University Journal of Science 26 4 677–686.
IEEE İ. Gökcan and A. H. Değer, “Investigation of Matrices Q^(n._L ), M^(n._L ) and R^(n._L ) and Some Related Identities”, SAUJS, vol. 26, no. 4, pp. 677–686, 2022, doi: 10.16984/saufenbilder.1034057.
ISNAD Gökcan, İbrahim - Değer, Ali Hikmet. “Investigation of Matrices Q^(n._L ), M^(n._L ) and R^(n._L ) and Some Related Identities”. Sakarya University Journal of Science 26/4 (August 2022), 677-686. https://doi.org/10.16984/saufenbilder.1034057.
JAMA Gökcan İ, Değer AH. Investigation of Matrices Q^(n._L ), M^(n._L ) and R^(n._L ) and Some Related Identities. SAUJS. 2022;26:677–686.
MLA Gökcan, İbrahim and Ali Hikmet Değer. “Investigation of Matrices Q^(n._L ), M^(n._L ) and R^(n._L ) and Some Related Identities”. Sakarya University Journal of Science, vol. 26, no. 4, 2022, pp. 677-86, doi:10.16984/saufenbilder.1034057.
Vancouver Gökcan İ, Değer AH. Investigation of Matrices Q^(n._L ), M^(n._L ) and R^(n._L ) and Some Related Identities. SAUJS. 2022;26(4):677-86.
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