**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

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### 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).

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## Cite this paper

Chung, C.-L., & Zhong, C. (2024). Melham’s sums for some Lucas polynomial sequences. *Notes on Number Theory and Discrete Mathematics*, 30(2), 383-409, DOI: 10.7546/nntdm.2024.30.2.383-409.