Some equalities and binomial sums about the generalized Fibonacci number u(n)

In this study, we take the generalized Fibonacci sequence {u(n)} as u(0) = 0; u(1) = 1 and u(n) = ru(n-1) + u(n-2) for n > 1, where r is a non-zero integer. Based on Halton's paper in [4], we derive three interrelated functions involving the terms of generalized Fibonacci sequence {u(n)}. Using these three functions we introduce a simple approach to obtain a lot of identities, binomial sums and alternate binomial sums involving the terms of generalized Fibonacci sequence {u(n)}.