Melham’s sums for some Lucas polynomial sequences

Chan-Liang Chung and Chunmei Zhong
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 2, Pages 383–409
DOI: 10.7546/nntdm.2024.30.2.383-409
Full paper (PDF, 295 Kb)

Details

Authors and affiliations

Chan-Liang Chung
School of Mathematics and Statistics, Fuzhou University
Fuzhou 350100, China

Chunmei Zhong
School of Mathematics and Statistics, Fuzhou University
Fuzhou 350100, China

Abstract

A Lucas polynomial sequence is a pair of generalized polynomial sequences that satisfy the Lucas recurrence relation. Special cases include Fibonacci polynomials, Lucas polynomials, and Balancing polynomials. We define the -type Lucas polynomial sequences and prove that their Melham’s sums have some interesting divisibility properties. Results in this paper generalize the original Melham’s conjectures.

Keywords

• Lucas polynomial sequence
• Fibonacci sequence
• Lucas sequence
• Melham’s conjectures

2020 Mathematics Subject Classification

• 11B39
• 05A19
• 11B37

References

1. Behera, A., & Panda, G. K. (1999). On the square roots of triangular numbers. The Fibonacci Quarterly, 37(2), 98–105.
2. Brewer, B. W. (1961). On certain character sums. Transactions of the American
   Mathematical Society, 99(2), 241–245.
3. Chen, L., Wang, X. (2019). The power sums involving Fibonacci polynomials and their applications. Symmetry, 11(5), Article 635.
4. Chung, C.-L. (2019). Some polynomial sequence relations. Mathematics, 7(8), Article 750.
5. Horadam, A. F. (1996). A synthesis of certain polynomial sequences. Applications of Fibonacci Numbers, 6, 215–229.
6. Jennings, D. (1993). Some polynomial identities for the Fibonacci and Lucas numbers. The Fibonacci Quarterly, 31(2), 134–137.
7. Leslie, R. A. (1991). How not to repeatedly differentiate a reciprocal. The American Mathematical Monthly, 98, 732–735.
8. Melham, R. S. (2009). Some conjectures concerning sums of odd powers of Fibonacci and Lucas numbers. The Fibonacci Quarterly, 46/47, 312–315.
9. Ozeki, K. (2008). On Melham’s sum. The Fibonacci Quarterly, 46, 107–110.
10. Patel, B. K., Irmak, N., & Ray, P. K. (2018). Incomplete balancing and Lucas-balancing numbers. Mathematical Reports, 20, 59–72.
11. Prodinger, H. (2008). On a sum of Melham and its variants. The Fibonacci Quarterly, 46, 207–215.
12. Sun, B. Y., Xie, M. H. Y., & Yang, A. L. B. (2016). Melham’s conjecture on odd power sums of Fibonacci numbers. Quaestiones Mathematicae, 39, 945–957.
13. Wang, T. T., & Zhang, W. P. (2012). Some identities involving Fibonacci, Lucas polynomials and their applications. Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, 55(103), 95–103.
14. Weisstein, E. W. Fermat Polynomial. From MathWorld–A Wolfram Web Resource. Available online at: http://mathworld.wolfram.com/FermatPolynomial.html.
15. Weisstein, E. W. Fermat–Lucas Polynomial. From MathWorld–A Wolfram
    Web Resource. Available online at: http://mathworld.wolfram.com/Fermat-LucasPolynomial.html.
16. Wiemann, M., & Cooper, C. (2004). Divisibility of an F−L type convolution. Applications of Fibonacci Numbers, 9, 267–287.

Manuscript history

• Received: 25 August 2023
• Revised: 16 May 2024
• Accepted: 28 May 2024
• Online First: 30 May 2024

Copyright information

Ⓒ 2024 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).