Some identities involving differences of products of generalized Fibonacci numbers

Curtis Cooper

Curtis Cooper

Colloquium Mathematicae, Tome 139 (2015), p. 45-49 / Harvested from The Polish Digital Mathematics Library

Résumé

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

Publié le : 2015-01-01

Zbl 06459963

EUDML-ID : urn:eudml:doc:283999

@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm141-1-4,
     author = {Curtis Cooper},
     title = {Some identities involving differences of products of generalized Fibonacci numbers},
     journal = {Colloquium Mathematicae},
     volume = {139},
     year = {2015},
     pages = {45-49},
     zbl = {06459963},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm141-1-4}
}

Curtis Cooper. Some identities involving differences of products of generalized Fibonacci numbers. Colloquium Mathematicae, Tome 139 (2015) pp. 45-49. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm141-1-4/