Yasemin Alp and E. Gökçen Koçer

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 1, Pages 167–177

DOI: 10.7546/nntdm.2019.25.1.167-177

**Full paper (PDF, 171 Kb)**

## Details

### Authors and affiliations

Yasemin Alp

*Department of Mathematics-Computer Sciences, Faculty of Science
University of Necmettin Erbakan, Meram-Konya, Turkey
*

E. Gökçen Koçer

*Department of Mathematics-Computer Sciences, Faculty of Science
University of Necmettin Erbakan, Meram-Konya, Turkey
*

### Abstract

In this paper, we consider the space *R*(*p*, 1) of generalized Fibonacci sequences and orthogonal bases of this space. Using these orthogonal bases, we obtain the orthogonal projection onto a subspace *R*(*p*, 1) of ℝ^{n}. By using the orthogonal projection, we obtain the identities for the generalized Fibonacci numbers.

### Keywords

- Fibonacci numbers
- Orthogonal basis

### 2010 Mathematics Subject Classification

- 11B39
- 15A03

### References

- Alp, Y., & Ko
*ç*er, E. G. (2018). The Pell, Modified Pell Identities via Orthogonal Projection, Journal of Applied Mathematics and Computation (JAMC), 2 (3), 67–78. - Bretscher, O. (2015). Linear Algebra with Applications, Fifth Edition, Pearson Education.
- Dupree, E., & Mathes, B. (2012). Singular Values of k-Fibonacci and
*k*-Lucas Hankel matrices, Int. J. Contemp. Math Sciences, 47 (7), 2327–2339. - Fraleigh, J. B., & Beauregard, R. A. (1995). Linear Algebra, Addison-Wesley Publishing Company.
- Hawkins, K., Hebert-Johnson, U., & Mathes, B. (2015). The Fibonacci Identities of Orthogonality, Linear Algebra and Its Application, 475, 80–89.
- Horadam, A. F. (1965). Basic Properties of a Certain Generalized Sequence of Numbers, The Fibonacci Quarterly, 3 (3), 161–176.
- Horzum, T., & Ko
*ç*er, E. G. (2009). On Some Properties of Horadam Polynomials, International Mathematical Forum, 4 (25), 1243–1252. - Kalman, D., & Mena, R. (2003). The Fibonacci Numbers–Exposed, Mathematics Magazine, 76 (3), 167–181.
- Mansour, T. (2004). A formula for the generating functions of powers of Horadam’s Sequence, Australasian Journal of Combinatorics, 30, 207–212.

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## Cite this paper

Alp, Y. & Koçer, E. G. (2019). The identities for generalized Fibonacci numbers via orthogonal projection. *Notes on Number Theory and Discrete Mathematics*, 25(1), 167-177, DOI: 10.7546/nntdm.2019.25.1.167-177.