Horadam-Lagrange Interpolation Polynomials: Construction, Recurrence Relations, and Connections to Special Number Sequences
DOI:
https://doi.org/10.61383/ejam.20253195Keywords:
Fibonacci numbers, Horadam numbers, Lagrange interpolation polynomialsAbstract
This study investigates the construction of polynomials of at most degree \(n\) using the first \(n+1\) terms of the Horadam sequence through Lagrange interpolation. The paper provides a comprehensive analysis of the recurrence relations and fundamental identities associated with the Horadam-Lagrange Interpolation Polynomials. Furthermore, it explores the structural properties and special cases of these polynomials, highlighting their connections to well-known sequences such as Fibonacci, Lucas, Pell, Jacobsthal, Mersenne and Fermat sequences.
References
[1] O. Dis¸kaya, H. Menken, On the Jacobsthal and Jacobsthal-Lucas Subscripts, J. Algebra Comput. Appl, 8 (2019), 1-6.
[2] J. Kiusalaas, Numerical methods in engineering with Python. Cambridge university press, 2010. DOI: https://doi.org/10.1017/CBO9780511812224
[3] M. S. U. Mufid, T. Asfihani, L. Hanafi, On the Lagrange interpolation of Fibonacci sequence, (IJCSAM) International Journal of Computing Science and Applied Mathematics, 2(2016), no.3, 38-40. DOI: https://doi.org/10.12962/j24775401.v2i3.2093
[4] B.Hopkins and A. Tangboonduangjit, Fibonacci-producing rational polynomials. The Fibonacci Quarterly, 56(2018), no. 4, 303-312. DOI: https://doi.org/10.1080/00150517.2018.12427677
[5] E. S ¨ uli, D. F. Mayers, An introduction to numerical analysis, Cambridge university press, 2003. DOI: https://doi.org/10.1017/CBO9780511801181
[6] O. Dis¸kaya, On the Lagrange Interpolations of the Jacobsthal and Jacobsthal-Lucas Sequences. Journal of Universal Mathematics, 7(To memory” Assoc. Prof. Dr. Zeynep Akdemirci S¸anlı”), (2024), 128-137. DOI: https://doi.org/10.33773/jum.1518403
[7] A. F. Horadam, A generalized Fibonacci sequence. The American Mathematical Monthly, 68(1961), no. 5, 455-459. DOI: https://doi.org/10.1080/00029890.1961.11989696
[8] A. F. Horadam, Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3 (1965), no. 3, 161-176. DOI: https://doi.org/10.1080/00150517.1965.12431416
[9] A. F. Horadam, Special properties of the sequence Wn(a, b; p, q). The Fibonacci Quarterly, 5(1967), no. 5, 424-434. DOI: https://doi.org/10.1080/00150517.1967.12431271
[10] T. Koshy, Pell and Pell–Lucas Numbers, Springer New York, 2014. DOI: https://doi.org/10.1007/978-1-4614-8489-9_7
[11] A. F. Horadam, Pell identities. The Fibonacci Quarterly, 9(1971), no. 3, 245-263. DOI: https://doi.org/10.1080/00150517.1971.12431004
[12] A. F. Horadam, Jacobsthal representation numbers. The Fibonacci Quarterly, 34(1996), no. 1, 40-54. DOI: https://doi.org/10.1080/00150517.1996.12429096
[13] R. M. Robinson, Mersenne and Fermat numbers. Proceedings of the American Mathematical Society, 5(1954), no. 5, 842-846. DOI: https://doi.org/10.1090/S0002-9939-1954-0064787-4
[14] N. N. Vorobiev, Fibonacci numbers. Springer Science & Business Media, 2002. DOI: https://doi.org/10.1007/978-3-0348-8107-4
[15] S. Vajda, Fibonacci and Lucas numbers, and the golden section: theory and applications. Courier Corporation, 2008.
[16] A. Dasdemir, On the Pell, Pell-Lucas and modified Pell numbers by matrix method. Applied Mathematical Sciences, 5(2011), no. 64, 3173-3181.
[17] H. Ozimamo˘ glu, On hyper complex numbers with higher order Pell numbers components. The Journal of Analysis, 31(2023), no. 4, 2443-2457. DOI: https://doi.org/10.1007/s41478-023-00579-2
[18] H. Ozimamo˘ glu, and A. Kaya, The Linear Algebra of the Pell-Lucas Matrix. Fundamental Journal of Mathematics and Applications, 7(2024), no.3, 158-168. DOI: https://doi.org/10.33401/fujma.1404456
[19] E. Kilic¸ and D. Tas¸ci, The generalized Binet formula, representation and sums of the generalized order-k Pell numbers. Taiwanese Journal of Mathematics, 10(2006), no.6, 1661-1670. DOI: https://doi.org/10.11650/twjm/1500404581
[20] C. B. C¸ imen and A. Ipek, On jacobsthal and jacobsthal–lucas octonions. Mediterranean Journal of Mathematics, 14(2017), 1-13. DOI: https://doi.org/10.1007/s00009-017-0873-2
[21] Y. Soykan, A study on generalized Mersenne numbers. Journal of Progressive Research in Mathematics, 18 (2021), no. 3, 90-112.
[22] E. Eser, B. Kulo˘ glu, and E. Ozkan, An Encoding-Decoding Algorithm Based On Fermat And Mersenne Numbers. Applied Mathematics E-Notes, 24(2024).
[23] R. Skurnick, A. Baderian, and J. Martin, A note on the extended binomial coefficient for negative integers and an extension of Pascal’s triangle, Math. Comput. Education, 46(2012), 35–40.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Orhan Diskaya

This work is licensed under a Creative Commons Attribution 4.0 International License.