On Sums Involving Polynomials and Generalized Fibonacci Sequences

Let $(s_n)_{n\geq 0}$ be a generalized Fibonacci sequence with initial values $s_0 = c_0$, $s_1 = c_1$ that satisfies the recurrence relation $s_{n+1}=as_{n}+bs_{n-1}$ for all positive integers $n$, where $a,b\in\mathbb N$, $c_0,c_1\in\mathbb Z$, $(c_0,c_1)\neq (0,0)$. In this paper, we get the result that for every polynomials $P(x)$ with real coefficients, we can always find three polynomials $F_1(x), G_1(x), H_1(x)$ with real coefficients satisfying the identity: $\;\sum_{k=1}^{n}P(k)s_{k-1} = F_1(n)s_{n+1} + G_1(n)s_n + H_1(n) \;$ for all positive integers $n$. Furthermore, we present two cases for $(s_n)_{n\geq 0}$: one case implies that there are infinitely many triples $(F_1(x), G_1(x), H_1(x))$ satisfying that identity, while another one implies that there is only one triple $(F_1(x), G_1(x), H_1(x))$ satisfying that identity.