Melham's Conjecture on Odd Power Sums of Fibonacci Numbers

TL;DR

This paper proves Melham's conjecture by showing that a specific polynomial related to odd power sums of Fibonacci numbers vanishes at 1 and becomes an integer polynomial after a certain multiplication, confirming a long-standing hypothesis.

Contribution

It establishes that the polynomial and its derivative vanish at 1 and becomes an integer polynomial after multiplication, confirming Melham's conjecture.

Findings

Polynomial and its derivative vanish at 1

Polynomial becomes integer after multiplication by Lucas numbers

Affirmative proof of Melham's conjecture

Abstract

Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at certain Fibonacci number of odd order. We prove that this polynomial and its derivative both vanish at $1$, and will be an integer polynomial after multiplying it by a product of the first consecutive Lucas numbers of odd order. This presents an affirmative answer to a conjecture of Melham.