Sums of powers of generalized Fibonacci numbers (Q2883420)

The sequence \(\{w_n\}\) defined by \(w_{n+2} = aw_{n+1} + bw_{n}\) with initial conditions \(w_0 = c_0\), \(w_1 = c_1\), where \(a, b, c_0, c_1\) are complex constants, is called a second-order Fibonacci-Lucas sequence, or simply, an F-L sequence. Its characteristic polynomial is defined by \(f(x) = x^2 - ax^{1} - b.\) Let \(\Delta = a^2+4b\).NEWLINENEWLINESome representations of expressions \(\Delta^k w_n^m\) for \(m = 2k\) or \(2k+1\), are formulated, proved and extended.