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

**Download full paper: PDF, 173 Kb**

## Details

### 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

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

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

## Related papers

## Cite this paper

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

ChicagoŞahin, Adem, and Makoto Minamide. “On an Analogue of Buchstab’s Identity.” Notes on Number Theory and Discrete Mathematics 22, no. 1 (2016): 8-17.

MLAŞahin, Adem, and Kenan Kaygısız. “On an Analogue of Buchstab’s Identity.” Notes on Number Theory and Discrete Mathematics 22.1 (2016): 8-17. Print.