Some identities involving differences of products of generalized Fibonacci numbers

Curtis Cooper. "Some identities involving differences of products of generalized Fibonacci numbers." Colloquium Mathematicae 141.1 (2015): 45-49. <http://eudml.org/doc/283999>.

@article{CurtisCooper2015,
abstract = {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 $WₙW_\{n+4\}W_\{n+5\} - W³_\{n+3\} = eqⁿ(p³W_\{n+4\} - qW_\{n+5\})$.},
author = {Curtis Cooper},
journal = {Colloquium Mathematicae},
keywords = {generalized Fibonacci numbers; Fibonacci identities; differences of products},
language = {eng},
number = {1},
pages = {45-49},
title = {Some identities involving differences of products of generalized Fibonacci numbers},
url = {http://eudml.org/doc/283999},
volume = {141},
year = {2015},
}

TY - JOUR
AU - Curtis Cooper
TI - Some identities involving differences of products of generalized Fibonacci numbers
JO - Colloquium Mathematicae
PY - 2015
VL - 141
IS - 1
SP - 45
EP - 49
AB - 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 $WₙW_{n+4}W_{n+5} - W³_{n+3} = eqⁿ(p³W_{n+4} - qW_{n+5})$.
LA - eng
KW - generalized Fibonacci numbers; Fibonacci identities; differences of products
UR - http://eudml.org/doc/283999
ER -