On Sum of a Polynomial Multiplied by Generalized Fibonacci Numbers

Given that $a,b\in\mathbb N$, $c_0,c_1\in\mathbb Z$, $(c_0,c_1)\neq (0,0)$, and a generalized Fibonacci sequence $(s_n)_{n\geq 0}$ where $s_0 = c_0$, $s_1 = c_1$, and $s_{n+1}=as_{n}+bs_{n-1}$ for all positive integers $n$. In this paper, we get the result that for every polynomials $P(x)$ with real coefficients, we can always find three polynomials $F_1(x), G_1(x), H_1(x)$ (not necessarily distinct) with real coefficients satisfying the identity: $\;2\sum_{k=1}^{n}P(k)s_{k-1} = F_1(n)s_{n+1} + G_1(n)s_n + H_1(n), \;\forall n\in\mathbb N$. Furthermore, we serve two constraints for $(s_n)_{n\geq 0}$: one constraint implies that there are infinitely many triples $(F_1(x), G_1(x), H_1(x))$ satisfying the identity $\;2\sum_{k=1}^{n}P(k)s_{k-1} = F_1(n)s_{n+1} + G_1(n)s_n + H_1(n), \;\forall n\in\mathbb N$, while another constraint implies that there is only one triple $(F_1(x), G_1(x), H_1(x))$ satisfying the identity $\;2\sum_{k=1}^{n}P(k)s_{k-1} = F_1(n)s_{n+1} + G_1(n)s_n + H_1(n), \;\forall n\in\mathbb N$.