Some identities involving differences of products of generalized Fibonacci numbers

• Autorzy: Curtis Cooper
• Języki publikacji: EN

Abstrakty

EN

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
$WₙW_{n+4}W_{n+5} - W³_{n+3} = eqⁿ(p³W_{n+4} - qW_{n+5})$.

Kategorie tematyczne

• 11B37: Recurrences
• 11B39: Fibonacci and Lucas numbers and polynomials and generalizations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2015
• Tom: 141
• Numer: 1
• Strony: 45-49

Daty

wydano

2015

Twórcy

autor

Curtis Cooper

• Department of Mathematics and Computer Science, University of Central Missouri, Warrensburg, MO 64093, U.S.A.

Identyfikatory

DOI

10.4064/cm141-1-4