Inverse of triangular matrices and generalized bivariate Fibonacci and Lucas p-polynomials

Adem Şahin and Kenan Kaygısız
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 22, 2016, Number 1, Pages 18–28
Full paper (PDF, 173 Kb)

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Authors and affiliations

Adem Şahin
Faculty of Education, Gaziosmanpaşa University
60250 Tokat, Turkey

Kenan Kaygısız
Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpaşa University
60250 Tokat, Turkey

Abstract

In this paper, we compute the generalized bivariate Fibonacci and Lucas p-polynomials by using inverse of various triangular matrices. In addition, in each calculation, instead of obtaining a type of sequence only, we are able to determine successive n terms of the two types of polynomial sequences simultaneously.

Keywords

  • Generalized bivariate Fibonacci and Lucas p-polynomials
  • Triangular matrix

AMS Classification

  • 11B37
  • 15A15
  • 15A51

References

  1. Chen, Y-H., & C-Y. Yu (2011) A new algorithm for computing the inverse and the determinant of a Hessenberg matrix, Appl. Math. Comput., 218, 4433-4436.
  2. Er, M. C. (1984) Sums of Fibonacci Numbers by Matrix Method, Fibonacci Quart. , 23(3), 204–207.
  3. Horadam, A. F., & J. M. Mahon, Br. (1985) Pell and Pell–Lucas Polynomials, Fibonacci Quart., 23(1), 17–20.
  4. Kaygısız, K. & A. Şahin (2011) Determinantal and permanental representation of generalized bivariate Fibonacci p-polynomials, (arXiv:1111.4071v1).
  5. Kaygısız, K. & A. Şahin (2012) Generalized bivariate Lucas p-polynomials and Hessenberg Matrices, J. Integer Seq., 15, Article 12.3.4.
  6. Kaygısız, K. & A. Şahin (2013) Determinants and Permanents of Hessenberg Matrices and Generalized Lucas Polynomials, Bull. Iranian Math. Soc., 39(6), 1065–1078
  7. Kaygısız, K. & A. Şahin (2013) A new method to compute the terms of generalized order-k Fibonacci numbers, J. Number Theory, 133, 3119–3126.
  8. Kaygısız, K. & A. Şahin (2016) Determinantal and Permanental Representations of Fibonacci Type Numbers and Polynomials, Rocky Mountain J. Math., to appear.
  9. Kaygısız, K. & A. Şahin (2014) Calculating terms of associated polynomials of Perrin and Cordonnier numbers. Notes on Number Theory and Discrete Mathematics, 20(1), 10–18.
  10. Kılıç, E., & A. P. Stakhov (2009) On the Fibonacci and Lucas p numbers, their sums, families of bipartite graphs and permanents of certain matrices, Chaos Solitions Fract., 40, 2210–2221.
  11. Kılıç, E., & D. Taşcı(2010) On the generalized Fibonacci and Pell sequences by Hessenberg matrices, Ars Combin., 94, 161–174.
  12. Lee, G.-Y., & S.-G. Lee (1995) A Note on Generalized Fibonacci Numbers, Fibonacci Quart., 33, 273–278.
  13. Lee, G.-Y. (2000) k-Lucas Numbers and Associated Bipartite Graphs,Lineer Algebra Appl.,320, 51–61.
  14. Lupas, A. (1999) A Guide of Fibonacci and Lucas Polynomial, Octagon Mathematics Magazine, 7, 2–12.
  15. MacHenry, T. (1999) A Subgroup of The Group of Units in The Ring of Arithmetic Fonctions, Rocky Mountain J. Math., 39, 1055–1065.
  16. MacHenry, T. (2000) Generalized Fibonacci and Lucas Polynomials and Multiplicative Arithmetic Functions, Fibonacci Quart., 38, 17–24.
  17. Minc, H. (1978) Encyclopaedia of Mathematics and its Applications, Permanents , Vol.6, Addison-Wesley Publishing Company, London.
  18. Tuglu, N., E.G. Kocer & A. Stakhov (2011) Bivariate fibonacci like p-polynomials, Appl. Math. Comput., 217, 10239–10246.
  19. Şahin, A. (2015) Matrix Representations of Weighted Isobaric Polynomials, UJMMS , 8(1-2), 59-73.
  20. Şahin, A. (2016) On the Q analogue of fibonacci and lucas matrices and fibonacci polynomials, Utilitas Mathematica, appear, Vol. 100.
  21. Şahin, A., & J. L. Ramirez (2016) Determinantal and permanental representations of convolved Lucas polynomials, Appl. Math. Comput., 281, 314–322.
  22. Udrea, G. (1998) Chebyshev Polynomials and Some Methods of Approximation, Port. Math., Fasc.3, 55, 261–269.

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Cite this paper

Şahin, A. & Kaygısız K. (2016). On an analogue of Buchstab’s identity. Notes on Number Theory and Discrete Mathematics, 22(1), 8-17.

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