Semi-analytical solutions for eigenvalue problems of chains and periodic graphs

We first show the existence and nature of convergence to a limiting set roots for polynomials in three-term recurrence form $p_{n+1}(z) = Q_k(z)p_{n}(z)+ \gamma p_{n-1}(z)$ as $n$ $\rightarrow$ $\infty$, where coefficient $Q_k(z)$ is $k^{th}$ degree polynomial, $z,\gamma \in \mathbb{C}$. extend these results relations numerically approximating such any given $n$. General solutions evaluation are motivated by large computational efforts errors iterative numerical methods. Later, we apply this solution eigenvalue problems represented tridiagonal matrices with periodicity $k$ its entries, providing more accurate method spectra chains reduction effort from $\mathcal{O}(n^2)$ $\mathcal{O}(n)$. also that along spectral rules Kronecker products allow an efficient many spatial lattices other periodic graphs.