ON THE LAGRANGE INTERPOLATIONS OF THE JACOBSTHAL AND JACOBSTHAL-LUCAS SEQUENCES

Abstract

This study explores the formation of polynomials of at most degree $n$ using the first $n+1$ terms of the Jacobsthal and Jacobsthal-Lucas sequences through Lagrange interpolation. The paper provides a detailed examination of the recurrence relations and various identities associated with the Jacobsthal and Jacobsthal-Lucas Lagrange Interpolation Polynomials.

Keywords

• Fibonacci numbers
• Lucas numbers
• Jacobsthal numbers
• Lagrange interpolations
• Binet formula

References

1. M. Asci, E. GUrel, Gaussian Jacobsthal and Gaussian Jacobsthal Lucas Numbers, Ars Combinatoria, Vol.5, No.111, pp.53-62 (2013).
2. P. Catarino, P. Vasco, H. Campos, A. P. Aires, A. Borges, New families of Jacobsthal and Jacobsthal-Lucas numbers, Algebra and Discrete Mathematics, vol.20, No.1, (2015).
3. G. Cerda-Morales, On bicomplex third-order Jacobsthal numbers, Complex Variables and Elliptic Equations, Vol.68, No.1, pp.44-56 (2023).
4. C. K. Cook, M. R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations. In Annales mathematicae et informaticae, pp. 27-39 (2013).
5. C. B. Cimen, A. Ipek, On jacobsthal and jacobsthal–lucas octonions, Mediterranean Journal of Mathematics, Vol.14, pp.1-13 (2017).
6. A. DaSdemir, On the Jacobsthal numbers by matrix method. SUleyman Demirel University Faculty of Arts and Science Journal of Science, vol.7, No.1, pp.69-76 (2012).
7. O. Deveci and G. Artun, On the adjacency-Jacobsthal numbers, Communications in Algebra, vol.47, No.11, pp.4520-4532 (2019).
8. C. M. Dikmen, Hyperbolic jacobsthal numbers, Asian Research Journal of Mathematics, Vol.15, No.4, pp.1-9 (2019). O. Diskaya, H. Menken, On the Jacobsthal and Jacobsthal-Lucas Subscripts, J. Algebra Comput. Appl, Vol.8, pp.1-6 (2019).

9. V. E. Hoggatt Jr, M. Bicknell-Johnson, Convolution arrays for Jacobsthal and Fibonacci polynomials, The Fibonacci Quarterly, Vol.16, No.5, pp.385-402 (1978).
10. A. F. Horadam, Jacobsthal representation numbers, significance, Vol.2, pp.2-8 (1996).
11. C. KızılateS, On the Quadra Lucas-Jacobsthal Numbers, Karaelmas Science Engineering Journal/Karaelmas Fen ve MUhendislik Dergisi, Vol.7, No.2, (2017).
12. J. Kiusalaas, Numerical methods in engineering with Python. Cambridge university press, (2010). E. Ozkan, B. Kuloglu, On a Jacobsthal-like sequence associated with k-Jacobsthal-Lucas sequence. Journal of Contemporary Applied Mathematics, Vol.10, No.2, pp. 3-13, (2020).
13. E. Ozkan, M. Uysal, B. Kuloglu, Catalan transform of the incomplete Jacobsthal numbers and incomplete generalized Jacobsthal polynomials. Asian-European Journal of Mathematics, Vol.15, No.06, pp. 2250119, (2022).
14. B. Kuloğlu, E. Ozkan, Applications of Jacobsthal and Jacobsthal-Lucas Numbers in Coding Theory, pp. 54-64, (2023).
15. M. S. U. Mufid, T. Asfihani, L. Hanafi, On the Lagrange interpolation of Fibonacci sequence, (IJCSAM) International Journal of Computing Science and Applied Mathematics, Vol.2, No.3, pp.38-40 (2016).
16. E. Ozkan, M. Uysal, On quaternions with higher order Jacobsthal numbers components, Gazi University Journal of Science, pp.1-1 (2023).
17. E. E. Polatlı, Y. Soykan On generalized third-order Jacobsthal numbers, Asian Research Journal of Mathematics, Vol.17, No.2, pp.1-19, (2021).
18. E. SUli, D. F. Mayers, An introduction to numerical analysis, Cambridge university press, (2003).
19. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, (A001045). https://oeis.org/A001045, (1973).
20. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, (A014551). https://oeis.org/A014551, (1973).
21. S. Uygun, The (s, t)-Jacobsthal and (s, t)-Jacobsthal Lucas sequences, Applied Mathematical Sciences, Vol.70, No.9, pp.3467-3476 (2015).