Let $\{a_n\}_{n=0}^\infty$,$\{b_n\}_{n=0}^\infty$,$\{c_n\}_{n=0}^\infty$ be three complex sequences and satisfy \begin{eqnarray*} &&\sum_{k=0}^2\alpha_ka_{n+k}=0,\\ &&\sum_{k=0}^4\beta_kb_{n+k}=0,\\ &&b_n^2=c_na_n-c_{n-1}a_{n+1} \end{eqnarray*} for any $n$, where $\alpha_i$ and $\beta_i$ are some constants satisfying $\alpha_i=\alpha_{2-i}$, $\beta_i=\beta_{4-i}$.
Let us assume $a_n$,$b_n$,$c_n$are not constant sequences and $\alpha_k$,$\beta_k$,$\gamma_k$ are not always zeros.
Please try to prove that there exist nonzero constants $\gamma_k$, $k=0,1,\cdots8$ so that $\sum_{k=0}^8\gamma_kc_{n+k}=0$ hold for any $n$.
I want to know how to solve the above problem, or what knowledge should be used?
Thanks!