Melham's conjecture on odd power sums of fibonacci numbers
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 a 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.
Mathematics Subject Classification (2010): 11B39, 05A19.
Keywords: Fibonacci numbers, Lucas numbers, Fibonacci polynomials, Lucas polynomials, Melham's conjecture, the Ozeki-Prodinger formula