Korean J. Math.  Vol 23, No 3 (2015)  pp.371-377
DOI: https://doi.org/10.11568/kjm.2015.23.3.371

Complex factorizations of the generalized Fibonacci sequences $\{q_n\}$

Sang Pyo Jun


In this note, we consider a generalized Fibonacci sequence $\{q_n\}$. Then give a connection between the sequence $\{q_n\}$ and the Chebyshev polynomials of the second kind $U_n(x)$. With the aid of factorization of Chebyshev polynomials of the second kind $U_n(x)$, we derive the complex factorizations of the sequence $\{q_n\}$.


generalized Fibonacci sequences, tridiagonal matrices, Chebyshev polynomials, complex factorization.

Subject classification



Funding for this paper was provided by Namseoul University.

Full Text:



N. D. Cahill, J. R. D’Errico, and J. P. Spence, Complex factorizations of the Fibonacci and Lucas numbers, The Fibonacci Quarterly 41 (1) (2003), 13–19. (Google Scholar)

M. Edson and O. Yayenie, A new generalization of Fibonacci sequence and extended Binet’s formula, Integer 9 (2009), 639–654. (Google Scholar)

Y. K. Gupta, Y. K. Panwar and O. Sikhwal, Generalized Fibonacci Sequences, Theoretical Mathematics and Applications 2 (2) (2012), 115–124. (Google Scholar)

Y. K. Gupta, M. Singh and O. Sikhwal, Generalized Fibonacci-Like Sequence Associated with Fibonacci and Lucas Sequences, Turkish Journal of Analysis and Number Theory 2 (6) (2014), 233–238. (Google Scholar)

A. F. Horadam, A generalized Fibonacci sequences, Amer. Math. Monthly 68 (1961), 455–459. (Google Scholar)

H.J. Hsiao, On factorization of Chebyshev’s polynomials of the first kind, Bulletin of the Institute of Mathematics Academia Sinica 12 (1) (1984), 89–94. (Google Scholar)

Y. H. Jang and S. P. Jun, Linearization of generalized Fibonacci sequences, Korean J. Math. 22 (2014) (3), 443–454. (Google Scholar)

D. Kalman and R. Mena, The Fibonacci numbers - Exposed. The Mathematical Magazine 2 (2002). (Google Scholar)

A. Oteles and M. Akbulak, Positive integer power of certain complex tridiagonal matrices, Applied Mathematics and Computation, 219 (21) (2013), 10448– 10455. (Google Scholar)

T.J. Rivlin, The Chebyshev Polynomials–From Approximation Theory to Algebra and Number Theory, Wiley-Interscience, John Wiley, (1990). (Google Scholar)

O. Yayenie, A note on generalized Fibonacci sequences, Applied Mathematics and Computation 217 (2011), 5603–5611. (Google Scholar)

H. Zhang and Z. Wu, On the reciprocal sums of the generalized Fibonacci sequences, Adv. Differ. Equ. (2013), Article ID 377 (2013). (Google Scholar)


  • There are currently no refbacks.

ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr