A Note On The Generalizations of Jacobthal and Jacobthal-Lucas Sequences

Yıl 2020, Cilt: 2 Sayı: 1, 28 - 43, 30.07.2020

Yashwant Panwar

Öz

In this paper, we present the generalization of Jacobsthal and Jacobsthal-Lucas sequences by the recurrence relations,,,,,,,,,,,,,,,,, with the initial conditions,,,,,,,,,,,,,,,,,,,,,, . We establish some of the interesting properties of involving them. Also we describe and derive sums, connection formulae and Generating function. We have used their Binet’s formula to derive the identities.

Anahtar Kelimeler

Generalized Jacobsthal sequence, Generalized Jacobsthal-Lucas sequence, Binet’s formula, Generating function

Kaynakça

• [1] Bilgici, G. (2014) New Generalizations of Fibonacci and Lucas Sequences. Applied Mathematical Sciences, 8(29): 1429-1437.
• [2] Falcon, S. (2001) On the k-Lucas Numbers. International Journal of Contemporary Mathematical Sciences, 6(21): 1039-1050.
• [3] Falcon, S., Plaza, A. (2007) On the k-Fibonacci Numbers. Chaos,Solitons and Fractals, 32(5): 1615-1624.
• [4] Gupta, V. K., Panwar, Y. K. (2012) Common Factors of Generalized Fibonacci, Jacobsthal and Jacobsthal-Lucas numbers. International Journal of Applied Mathematical Research, 1(4): 377-382.
• [5] Gupta, V. K., Panwar, Y. K., Sikhwal, O. (2012) Generalized Fibonacci Sequences. Theoretical Mathematics & Applications, 2(2): 115-124.
• [6] Horadam, A. F., (1961) A Generalized Fibonacci Sequence. American Mathematical Monthly, 68(5): 455-459.
• [7] Horadam, A. F., (1996) Jacobsthal representation numbers. The Fibonacci Quarterly, 34(1): 40-54.
• [8] Kalman, D., Mena, R. (2002) The Fibonacci Numbers–Exposed. The Mathematical Magazine, 2.
• [9] Koshy, T. (2001) Fibonacci and Lucas numbers with applications. New York, Wiley- Interscience.
• [10] Panwar, Y. K., Rathore, G. P. S., Chawla, R. (2014) On the k-Fibonacci-like numbers. Turkish J. Anal. Number Theory, 2(1): 9-12.
• [11] Panwar, Y. K., Singh, B., Gupta, V. K. (2013) Generalized Identities Involving Common Factors of Generalized Fibonacci, Jacobsthal and Jacobsthal-Lucas numbers. International journal of Analysis and Application, 3(1): 53-59.
• [12] Panwar, Y. K., Singh, B., Gupta, V. K. (2013) Identities Involving Common Factors of Generalized Fibonacci, Jacobsthal and Jacobsthal-Lucas numbers. Applied Mathematics and and Physic, 1(4): 126-128.
• [13] Singh, B., Bhadouria, P., Sikhwal, O. (2013) Generalized Identities Involving Common Factors of Fibonacci and Lucas Numbers. International Journal of Algebra, 5(13): 637-645.
• [14] Singh, B., Sikhwal, O., Bhatnagar, S. (2010) Fibonacci-Like Sequence and its Properties. Int. J. Contemp. Math. Sciences, 5(18): 859-868.
• [15] Singh, B., Sisodiya, K., Ahmed, F. (2014) On the Products of k-Fibonacci Numbers and k-Lucas Numbers. International Journal of Mathematics and Mathematical Sciences. Article ID 505798, 4 pages. http://dx.doi.org/10.1155/2014/505798
• [16] Suvarnamani, A., Tatong, M. (2015) Some Properties of (𝑝, 𝑞)-Fibonnacci Numbers. Science and Technology RMUTT Journal, 5(2): 17-21.
• [17] Suvarnamani, A., Tatong, M. (2016) Some Properties of (𝑝, 𝑞)-Lucas Numbers. Kyungpook Mathematical Journal, 56(2): 367-370.
• [18] Taşyurdu, Y. (2019) Generalized (𝑝, 𝑞)-Fibonacci-Like Sequences and Their Properties. Journal of Research, 11(6): 43-52.
• [19] Taşyurdu, Y., Cobanoğlu, N., Dilmen, Z. (2016) On The a New Family of k-Fibonacci Numbers. Erzincan University Journal of Science and Thechnology, 9(1): 95-101.
• [20] Thongmoon, M. (2009) Identities for the common factors of Fibonacci and Lucas numbers. International Mathematical Forum, 4(7): 303–308.
• [21] Thongmoon, M. (2009) New identities for the even and odd Fibonacci and Lucas numbers. International Journal of Contemporary Mathematical Sciences, 4(14): 671–676.
• [22] Uygun, S., Owusu, E. (2016) A new generalization of Jacobsthal numbers (Bi-Periodic Jacobsthal Sequences). Journal of Mathematical Analysis, 7(4): 28-39.
• [23] Uygun, S., Owusu, E. (2019) A new generalization of Jacobsthal numbers (Bi-Periodic Jacobsthal Lucas Sequences). Journal of Advances in Mathematics and Computer Science, 34(5): 1-13.
• [24] Wani, A. A., Catarino, P., Rafiq, R. U. (2018) On the Properties of k-Fibonacci-Like Sequence. International Journal of Mathematics And its Applications, 6(1-A): 187-198.
• [25] Wani, A. A., Rathore, G. P. S., Sisodiya, K. (2016) On The Properties of Fibonacci-Like Sequence. International Journal of Mathematics Trends and Technology, 29(2): 80-86.
• [26] Yagmur, T. (2019) New Approach to Pell and Pell-Lucas Sequence. Kyungpook Mathematical Journal, 59(1): 23-34.