Bi-Periodic (p,q)-Fibonacci and Bi-Periodic (p,q)-Lucas Sequences

Year 2023, Volume 27, Issue 1, 1 - 13, 28.02.2023

Yasemin TAŞYURDU Naime Şeyda TÜRKOĞLU

https://doi.org/10.16984/saufenbilder.1148618

Abstract

In this paper, we define bi-periodic (p,q)-Fibonacci and bi-periodic (p,q)-Lucas sequences, which generalize Fibonacci type, Lucas type, bi-periodic Fibonacci type and bi-periodic Lucas type sequences, using recurrence relations of (p,q)-Fibonacci and (p,q)-Lucas sequences. Generating functions and Binet formulas that allow us to calculate the nth terms of these sequences are given and the convergence properties of their consecutive terms are examined. Also, we prove some fundamental identities of bi-periodic (p,q)-Fibonacci and bi-periodic (p,q)-Lucas sequences conform to the well-known properties of Fibonacci and Lucas sequences.

Keywords

Bi-periodic Fibonacci Numbers, Fibonacci Number, Generalized Fibonacci Numbers, Bi-periodic Lucas Numbers, Lucas Number

References

• [1] A. F. Horadam, “A Generalized Fibonacci Sequence,” The American Mathematical Monthly, vol. 68, no. 5, pp. 455-459, 1961.
• [2] S. Falcon, A. Plaza, “On the Fibonacci k-Numbers,” Chaos, Solitons & Fractals, vol. 32, no. 5, pp. 1615-24, 2007.
• [3] S. Falcon, “On the k-Lucas Numbers,” International Journal of Contemporary Mathematical Sciences, vol. 6, no. 21, pp. 1039-1050, 2011.
• [4] T. Koshy, “Fibonacci and Lucas Numbers with Applications,” vol. 1, 2nd Edition, Wiley-Interscience Publications, New York, 2017, 704p.
• [5] Y. Taşyurdu, N. Çobanoğlu, Z. Dilmen, “On the a New Family of k-Fibonacci Numbers,” Erzincan University Journal of Science and Technology, vol. 9 no. 1, pp. 95-101, 2016.
• [6] Y. K. Panwar, “A Note on the Generalized k-Fibonacci Sequence,” MTU Journal of Engineering and Natural Sciences, vol. 2, no. 2, pp. 29-39, 2021.
• [7] O. Deveci, Y. Aküzüm, “The Recurrence Sequences via Hurwitz Matrices,” Annals of the Alexandru Ioan Cuza University-Mathematics, vol. 63, no. 3, pp. 1-13, 2017.
• [8] A. F. Horadam, “Basic Properties of a Certain Generalized Sequence of Numbers,” Fibonacci Quarterly, vol. 3, no. 3, 161–176, 1965.
• [9] A. Suvarnamani, M. Tatong, “Some Properties of (p,q)-Fibonacci Numbers,” Science and Technology RMUTT Journal, vol. 5, no. 2, pp. 17-21, 2015.
• [10] A. Suvarnamani, “Some Properties of (p,q)-Lucas Number,” Kyungpook Mathematical Journal, vol. 56, pp. 367-370, 2016.
• [11] Y. Taşyurdu, “Generalized (p,q)-Fibonacci-Like Sequences and Their Properties,” Journal of Mathematics Research, vol. 11, no. 6, pp. 43-52, 2019.
• [12] M. Edson, O. Yayenie, “A New Generalization of Fibonacci Sequence & Extended Binet’s Formula,” Integers, vol. 9, pp. 639–654, 2009.
• [13] G. Bilgici, “Two Generalizations of Lucas Sequence,” Applied Mathematics and Computation, vol. 245, pp. 526–538, 2014.
• [14] O. Yayenie, “A Note on Generalized Fibonacci Sequence,” Applied Mathematics and Computation, vol. 217, pp. 5603-5611, 2011.
• [15] S. P. Jun, K. H. Choi, “Some Properties of the Generalized Fibonacci Sequence {q_n } by Matrix Methods,” Korean Journal Mathematics., vol. 24, no. 4, pp. 681-691, 2016.
• [16] E. Tan, “Some Properties of the bi-Periodic Horadam Sequences,” Notes on Number Theory and Discrete Mathematics, vol. 23, no. 4, pp. 56-65, 2017.
• [17] Ş. Uygun, E. Owusu, “A New Generalization of Jacobsthal Numbers (Bi-Periodic Jacobsthal Sequence),” Journal of Mathematical Analysis, vol. 7 no. 4, pp. 28-39, 2016.
• [18] Ş. Uygun, E. Owusu, “A New Generalization of Jacobsthal Lucas Numbers (Bi-Periodic Jacobsthal Lucas Sequence),” Journal of Advances in Mathematics and Computer Science, vol. 34, no. 5, pp. 1-13, 2019.
• [19] Ş. Uygun, H. Karatas, “Bi-Periodic Pell Sequence,” Academic Journal of Applied Mathematical Sciences, vol. 6 no. 7, pp. 136-144, 2020.
• [20] Ş. Uygun, H. Karatas, “A New Generalization of Pell-Lucas Numbers (Bi-Periodic Pell-Lucas Sequence),” Communications in Mathematics and Applications, vol. 10, no. 3, pp. 469-479, 2019.