SOME IDENTITIES INVOLVING DIFFERENCES OF PRODUCTS OF GENERALIZED FIBONACCI NUMBERS

Melham discovered the Fibonacci identity Fn+1Fn+2Fn+6 - F-n+3(3) = (-1)F-n(n). He then considered the generalized sequence W-n, where W-0 - a, W-1 - b, and W-n - pW(n-1) + qW(n-2) and a, b, p and q are integers and q not equal 0. Letting e = pab - qa(2) - b(2), he proved the following identity: Wn+1Wn+2Wn+6 -W-n+3(3) = eq(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: FnFn+4Fn+5-Fn+33 = (-1)(n+1) Fn+6. We prove similar identities. For example, a generalization of Fairgrieve and Gould's identity is WnWn+4Wn+5 -W-n+3(3) = eq(n) ((PWn+4)-W-3-qW(n+5)).