What are the eigenvalues and eigenvectors of a symmetric pentadiagonal Toeplitz matrix?

Question

\begin{equation} \begin{pmatrix} \alpha & \beta & \gamma & \dots & 0 & 0 & 0 \\ \beta & \alpha & \beta & \dots & 0 & 0 & 0 \\ \gamma & \beta & \alpha & \dots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots\\ 0 & 0 & 0 & \dots & \alpha & \beta & \gamma \\ 0 & 0 & 0 & \dots & \beta & \alpha & \beta \\ 0 & 0 & 0 & \dots & \gamma & \beta & \alpha \end{pmatrix} \end{equation}

For symmetric tridiagonal Toeplitz matrix ($\gamma=0$), we have a closed-form solution.

Is there a closed-form solution for the symmetric pentadiagonal toeplitz matrix?

Answer 1

There are no closed-form formulas in general. You can, for example, read https://arxiv.org/abs/1710.05243 and https://arxiv.org/abs/1903.10551

Answer 2

No, but it is possible to derive a generating function that gives the sum of the rows as the matrix is raised to some power $y$, for the $n \times n$ matrix. Summing the first $y$ values of the function (with coefficients in the series having the form $a_y z^y$ ) gives the sum. The pentadiagonal has an analytic solution because the cubic is solvable.