In this paper, the authors derive some algebraic relations on the generating functions for generalized Fibonacci sequence \(\{Q_{n}\}_{n=0}^{\infty }\) defined by \(Q_{0}=0,Q_{1}=1,Q_{m}=a_{j}Q_{m-1}+b_{j}Q_{m-2},m\equiv j\pmod k\), where \(k\geq 3\) is a fixed integer and \(a_{0},a_{1},\cdots,a_{k-1},b_{0},b_{1},\cdots,b_{k-1}\) are \(2k\) given real or complex numbers, with \(b_{j}\neq 0\) for \(0\leq j\leq k-1\).