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Some identities involving differences of products of generalized Fibonacci numbers

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Cooper C.

Colloquium Mathematicum

2015

tom 141

nr 1

45-49

Melham discovered the Fibonacci identity $F_{n+1}F_{n+2}F_{n+6} - F³_{n+3} = (-1)ⁿFₙ$. He then considered the generalized sequence Wₙ where W₀ = a, W₁ = b, and $Wₙ = pW_{n-1} + qW_{n-2}$ and a, b, p and q are integers and q ≠ 0. Letting e = pab - qa² - b², he proved the following identity: $W_{n+1}W_{n+2}W_{n+6} - W³_{n+3} = eq^{n+1}(p³W_{n+2} - q²W_{n+1})$. There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: $FₙF_{n+4}F_{n+5} - F³_{n+3} = (-1)^{n+1}F_{n+6}$. We prove similar identities. For example, a generalization of Fairgrieve and Gould's identity is $WₙW_{n+4}W_{n+5} - W³_{n+3} = eqⁿ(p³W_{n+4} - qW_{n+5})$.

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1 Colloquium Mathematicum

1 Cooper C.

1 2015

Wyniki wyszukiwania

Some identities involving differences of products of generalized Fibonacci numbers