Some identities involving differences of products of generalized Fibonacci numbers
Curtis Cooper (2015)
Colloquium Mathematicae
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Melham discovered the Fibonacci identity . He then considered the generalized sequence Wₙ where W₀ = a, W₁ = b, and and a, b, p and q are integers and q ≠ 0. Letting e = pab - qa² - b², he proved the following identity: . There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: . We prove similar identities. For example, a generalization of Fairgrieve and Gould’s identity is .