Binomial sums with k-Jacobsthal and k-Jacobsthal–Lucas numbers

A. D. Godase
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 3, Pages 466–476
DOI: 10.7546/nntdm.2022.28.3.466-476
Full paper (PDF, 213 Kb)

Details

Authors and affiliations

A. D. Godase
Department of Mathematics, V. P. College Vaijapur
Aurangabad (MH), India

Abstract

In this paper, we derive some important identities involving k-Jacobsthal and k-Jacobsthal–Lucas numbers. Moreover, we use multinomial theorem to obtain distinct binomial sums of k-Jacobsthal and k-Jacobsthal–Lucas numbers.

Keywords

• Jacobsthal number
• Jacobsthal–Lucas number
• k-Jacobsthal number
• k-Jacobsthal–Lucas number

2020 Mathematics Subject Classification

• 11B37
• 11B50

References

1. Campos, H., Catarino, P., Aires, A. P., Vasco, P., & Borges, A. (2014). On Some Identities of k-Jacobsthal–Lucas Numbers. International Journal of Mathematical Analysis, 8(10), 489–494.
2. Carlitz, L., & Ferns, H. (1970). Some Fibonacci and Lucas Identities. The Fibonacci Quarterly, 8(1), 61–73.
3. Cerin, Z. (2007). Sums of squares and products of Jacobsthal numbers. Journal of Integer Sequences, 10, Article 07.2.5.
4. Godase, A. D. (2022). Some Binomial Sums of k-Jacobsthal and k-Jacobsthal–Lucas numbers. Communications in Mathematics and Applications, submitted (2022).
5. Horadam, A. F. (1996). Jacobsthal Representation Numbers. The Fibonacci Quarterly, 34, 40–54.
6. Jhala, D., Rathore, G. P. S., & Sisodiya, K. (2014). Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes. Turkish Journal of Analysis and Number Theory, 2(4), 119–124.
7. Jhala, D., Rathore, G. P. S., & Sisodiya, K. (2013). On Some Identities for k-Jacobsthal Numbers. International Journal of Mathematical Analysis, 7(12), 551–556.
8. Koken, F., & Bozkurt, D. (2008). On the Jacobsthal–Lucas numbers by matrix methods. International Journal of Contemporary Mathematical Sciences, 3(33), 1629–1633.
9. Srisawat, S., Sriprad, W., & Sthityanak, O. (2015). On the k-Jacobsthal Numbers by Matrix Methods. Progress in Applied Science and Technology, 5(1), 70–76.
10. Uygun, S. (2015). The (s, t)-Jacobsthal and (s, t)-Jacobsthal Lucas Sequences. Applied Mathematical Sciences, 70(09), 3467–3476.
11. Uygun, S., & Eldogan, H. (2016). k-Jacobsthal and k-Jacobsthal Lucas Matrix Sequences. International Mathematical Forum, 11, 145–154.
12. Uygun, S., & Eldogan, H. (2016). Properties of k-Jacobsthal and k-Jacobsthal Lucas Sequences. General Mathematics Notes, 36(1), 34–47.
13. Uygun, S., & Owusu, E. (2016). A new generalization of Jacobsthal numbers (bi-periodic Jacobsthal sequences). Journal of Mathematical Analysis, 7(5), 28–39.
14. Uygun, S., & Uslu, K. (2016). The (s, t)-Generalized Jacobsthal Matrix Sequences.
    Computational Analysis, Springer Proceedings in Mathematics & Statistics, Vol. 155, 325–336.
15. Zhang, Z. (1997). Some Identities Involving Generalized Second-order Integer Sequences. The Fibonacci Quarterly, 35(3), 265–267.

Manuscript history

• Received: 22 February 2022
• Revised: 29 July 2022
• Accepted: 1 August 2022
• Online First: 2 August 2022