FIBONACCI AND LUCAS SEQUENCES AT NEGATIVE INDICES

Year 2016, Volume: 4 Issue: 1, 172 - 178, 01.04.2016

Serpil Halıcı , Zeynep Akyüz

Abstract

This study investigate the Fibonacci and Lucas sequences at neg- ative indices. In this paper we give the formulas of F􀀀(nk+r) and L􀀀(nk+r) depending on whether the indices are odd or even. For this purpose we con- sider a special matrix and we give various combinatorial identities related with the Fibonacci and Lucas sequences by using the matrix method. Some of the resulting identities are well known identities in the literature, but some of these are new.

Keywords

Horadam Sequence, Recurrence Relations, Matrix Methods

References

• [1] Akyuz, Z. Halici, S., On Some Combinatorial Identities Involving The Terms of Generalized Fibonacci and Lucas Sequences, Hacettepe Journal of Math. And Statistics 42(4), 431-435, 2013.
• [2] Freitag, Herta. On Summations and Expansions of Fibonacci Numbers, The Fibonacci Quar- terly, 11(1), 63-71, 1973.
• [3] Halici, S., Akyuz, Z.., Some Identities Deriving From the nth Power of Special Matrix, Advances in Dierence Equations. doi:10.1186/1687-1847-2012-223, 2012.
• [4] Koken, F. Bozkurt, D.,On Lucas Numbers by The Matrix Method, Hacettepe Journal of Mathematics and Statistics, 39(4), 471-475, 2010.
• [5] Koshy, T., Fibonacci and Lucas Numbers With Applications, A. Wiley-Interscience Publica- tion, 2001.
• [6] Latushkin, Yaroslav, and Vladimir Ushakov. A representation of regular subsequences of recurrent sequences, Fibonacci Quart. 43(1), 70-84, 2005.
• [7] Laughlin, J.,Combinatorial Identities Deriving From the Power of a Matrix, Integer : Elec- tronic J. of Combinatorial Number Theory 4, 1-15, 2004.
• [8] Laughlin, J.,Further Combinatorial Identities Deriving From the Power of a Matrix, Discrete Applied Mathematics, 154 , 1301-1308, 2006.
• [9] Mansour, Touk. Generalizations of some identities involving the Fibonacci numbers, arXiv preprint math/0301157, 2003.
• [10] Melham, R. S , Shannon A. G. Some Summation Identities Using Generalized Q -Matrices, The Fibonacci Quarterly, 33(1), 64-73, 1995.
• [11] Vajda, S. Fibonacci, Lucas numbers, and the golden section, Theory and Applications. Ellis Horwood Limited; 1989.
• [12] Zhang, Wenpeng. Some identities involving the Fibonacci numbers and Lucas numbers, Fibonacci Quart., 42, 149-154, 2004.