Let $\alpha$ be a (strictly) positive real number. Consider the following tridiagonal Toeplitz matrix $$ A=\alpha\begin{bmatrix} 0 & 1 & 0 &\cdots & 0\\ 1 & 0 & 1 &\ddots & \vdots \\ 0 & 1 & 0 & \ddots & 0\\ \vdots & \ddots & \ddots & \ddots & 1 \\ 0 & \cdots & 0 & 1 & 0 \end{bmatrix}. $$
My question. Does there exist a closed-form expression for $\exp(A)$?
I played around a little bit with the truncated series $\sum_{k=0}^N \frac{A^k}{k!}$ but I didn't manage to provide an answer to my question. Pointers to the literature are also welcome!
Hint (too long for a comment): tridiagonal Toeplitz matrices are known to have distinct eigenvalues, which can be explicitly calculated (see e.g. here and here). For the matrix in question, for example, the eigenvalues are $\lambda_k=2 \alpha \cos\left(\cfrac{k \pi}{n+1}\right)\,$, $k=1,2,\cdots,n$.
The matrix is therefore diagonalizable, and since the eigenvectors can also be explicitly calculated, it is possible to determine the invertible matrix $P$ and diagonal matrix $D$ such that $A=P\,D\,P^{-1}$.
Given that $\,A^n=P\,D^n\,P^{-1}\,$ it follows that $\,e^A=P\,e^D\,P^{-1}\,$ where $e^D$ is the diagonal matrix with $e^{\lambda_k}$ on the diagonal.