On Sum of a Polynomial Multiplied by Generalized Fibonacci Numbers

Given that $a,b\in \mathbb{N}$, $c_0,c_1\in \mathbb{Z}$, $(c_0,c_1)\mathrm{\ne }(0,0)$, and a generalized Fibonacci sequence $(s_n{)}_{n\ge 0}$ where $s_0=c_0$, $s_1=c_1$, and $s_{n+1}=a{s}_n+b{s}_{n-1}$ 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: $2{\sum }_{k=1}^{n}P(k)s_{k-1}=F_1(n)s_{n+1}+G_1(n)s_n+H_1(n),\mathrm{\forall }n\in \mathbb{N}$. Furthermore, we serve two constraints for $(s_n{)}_{n\ge 0}$: one constraint implies that there are infinitely many triples $(F_1(x),G_1(x),H_1(x))$ satisfying the identity $2{\sum }_{k=1}^{n}P(k)s_{k-1}=F_1(n)s_{n+1}+G_1(n)s_n+H_1(n),\mathrm{\forall }n\in \mathbb{N}$, while another constraint implies that there is only one triple $(F_1(x),G_1(x),H_1(x))$ satisfying the identity $2{\sum }_{k=1}^{n}P(k)s_{k-1}=F_1(n)s_{n+1}+G_1(n)s_n+H_1(n),\mathrm{\forall }n\in \mathbb{N}$.