Some identities involving differences of products of generalized Fibonacci numbers
Volume 141 / 2015
Abstract
Melham discovered the Fibonacci identity $$ F_{n+1} F_{n+2} F_{n+6} - F_{n+3}^3 = (-1)^n F_n . $$ He then considered the generalized sequence $W_n$ where $W_0 = a$, $W_1 = b$, and $W_n = p W_{n-1} + q W_{n-2}$ and $a$, $b$, $p$ and $q$ are integers and $q \not =0$. Letting $e = pab - qa^2 - b^2$, he proved the following identity: $$ W_{n+1} W_{n+2} W_{n+6} - W_{n+3}^3 = e q^{n+1} ( p^3 W_{n+2} - q^2 W_{n+1} ) . $$ There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: $$ F_n F_{n+4} F_{n+5} - F_{n+3}^3 = (-1)^{n+1} F_{n+6}. $$ We prove similar identities. For example, a generalization of Fairgrieve and Gould's identity is $$ W_n W_{n+4} W_{n+5} - W_{n+3}^3 = eq^n ( p^3 W_{n+4} - q W_{n+5} ). $$
