A Note On The Generalized k-Fibonacci Sequence

Year 2021, , 29 - 39, 18.12.2021

Yashwant Panwar

https://doi.org/10.46572/naturengs.937010

Cited By: 7

Abstract

In this paper, we present a generalization of well-known k-Fibonacci sequence. Namely, we defined generalized k-Fibonacci sequence. This sequence generalizes others, k-Fibonacci sequence, classical Fibonacci sequence, Pell sequence and Jacobsthal sequence. We establish some of the interesting properties of generalized k-Fibonacci sequence. Also, we obtain a generating function for them.

Keywords

Generalized k-Fibonacci sequence, k-Fibonacci sequence, Fibonacci sequence, Binet’s formula, Generating function

References

• Bilgici, G. (2014). New Generalizations of Fibonacci and Lucas Sequences, Applied Mathematical Sciences, 8(29): 1429-1437.
• Falcon, S., and Plaza, A. (2007). On the k-Fibonacci Numbers, Chaos,Solitons and Fractals, 32(5): 1615-1624. https://doi.org/10.1016/j.chaos.2006.09.022
• Falcon, S. (2011). On the k-Lucas Numbers, International Journal of Contemporary Mathematical Sciences, 6(21): 1039-1050.
• Falcon, S., and Plaza, A. (2008). The k-Fibonacci hyperbolic functions, Chaos,Solitons and Fractals, 38(2): 409-420.
• Falcon, S., and Plaza, A. (2007). The k-Fibonacci sequence and the Pascal 2-triangle, Chaos,Solitons and Fractals, 33(1): 38-49.
• Gupta, V. K., Panwar, Y. K., and Sikhwal, O. (2012) Generalized Fibonacci Sequences, Theoretical Mathematics & Applications, 2(2): 115-124.
• Horadam, A.F. (1961). A Generalized Fibonacci Sequence, American Mathematical Monthly, 68(5): 455-459. https://doi.org/10.1080/00029890.1961.11989696
• Horadam, A. F. (1996). Jacobsthal Representation Numbers. The Fib. Quart, 34(1): 40-54.
• Horadam, A. F. (1971). Pell Identities, The Fibonacci Quarterly, 9(3), 245-252.
• Kalman, D., and Mena, R. (2002). The Fibonacci Numbers–Exposed, The Mathematical Magazine, 2.
• Koshy, T. (2001). Fibonacci and Lucas numbers with applications, New York, Wiley-Interscience. https://doi.org/10.1002/9781118033067
• Panwar, Y. K., Rathore, G. P. S., and Chawla, R. (2014). On the k-Fibonacci-like numbers, Turkish J. Anal. Number Theory, 2(1): 9-12. https://doi.org/10.12691/tjant-2-1-3
• Singh, B., Sikhwal, O., and Bhatnagar, S. (2010). Fibonacci-Like Sequence and its Properties, Int. J. Contemp. Math. Sciences, 5(18): 859-868.
• Spivey, M.Z. (2006). Fibonacci Identities via the determinant sum Property, The College Mathematics Journal, 37(2): 286-289.
• Suvarnamani, A., and Tatong, M. (2015). Some Properties of (p,q)-Fibonnacci Numbers, Science and Technology RMUTT Journal, 5(2): 17-21.
• Suvarnamani, A., and Tatong, M. (2016). Some Properties of (p,q)-Lucas Numbers, Kyungpook Mathematical Journal, 56(2): 367-370. https://doi.org/10.5666/KMJ.2016.56.2.367
• Taşyurdu, Y. (2019). Generalized (p,q)-Fibonacci-Like Sequences and Their Properties, Journal of Mathematics Research, 11(6): 43-52. https://doi.org/10.5539/jmr.v11n6p43
• Taşyurdu, Y., Cobanoğlu, N., and Dilmen, Z. (2016). On The a New Family of k-Fibonacci Numbers, Erzincan University Journal of Science and Thechnology, 9(1): 95-101. https://doi.org/10.18185/eufbed.01209
• Vajda, S. (1989). Fibonacci and Lucas numbers, and the golden section. Theory and applications, Chichester: Ellis Horwood limited .
• Wani, A. A., Catarino, P., and Rafiq, R. U. (2018) On the Properties of k-Fibonacci-Like Sequence, International Journal of Mathematics And its Applications, 6(1-A): 187-198.
• Wani, A. A., Rathore, G. P. S., and Sisodiya, K. (2016). On The Properties of Fibonacci-Like Sequence, International Journal of Mathematics Trends and Technology, 29(2), 80-86. https://doi.org/10.14445/22315373/IJMTT-V29P51
• Yazlik, Y. aand Taskara, N. (2012). A note on generalized k-Horadam sequence, Computers and Mathematics with Applications, 63(1): 36-41.