Melham's Conjecture on Odd Power Sums of Fibonacci Numbers
QUAESTIONES MATHEMATICAE(2016)
摘要
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.
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关键词
Fibonacci numbers,Lucas numbers,Fibonacci polynomials,Lucas polynomials,Melham's conjecture,the Ozeki-Prodinger formula
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