A note on alternating weighted sums of Fibonacci numbers

In this note, we introduce a new convolution identity for the difference between classical Fibonacci numbers Fk and the alternating powers of k. We then apply this identity to develop a direct formula for the alternating convolution of Fibonacci numbers Fk and powers of k. Finally, we obtain an explicit formula for the alternating weighted Brousseau sums of the Fibonacci numbers using this alternating convolution identity. Furthermore, we generalize these identities to the Gibonacci numbers Gk with arbitrary initial values.