# Exponential of a symmetric 2-tridiagonal matrix

Is there any way of computing analytically the exponential (or finding the eigenvalues, or determining if it's positive definite) of a $n\times n$ symmetric 2-tridiagonal matrix: $$T_n^{(2)}= \begin{bmatrix} a_1 & 0 & b_1 & & & \\ 0 & a_2 & 0 & b_2 & & \\ b_1 & 0 & a_3 & 0 & b_3 & \\ & b_2 & 0 & a_4 & 0 & \ddots &\\ & & b_3 & 0 & a_5 & \ddots & b_{n-2}\\ & & & \ddots & \ddots & \ddots & 0 \\ & & & & b_{n-2} & 0 & a_n \end{bmatrix}$$

Actually I am interested only on the first and the second row of the $\exp[T_n^{(2)}]$, if that helps.

A partial result: If you permute the indices of this matrix as $$\{1,2,3,4,\cdots\}\mapsto\{1,3,5,\cdots,2,4,6,\cdots\}$$ then the matrix becomes block diagonal
$$T_n^{(2)}=\left( \begin{array}{cccc|cccc} a_1 & b_1 & 0&\cdots & \\ b_1 & a_3 & b_3 & \cdots & & \ddots \\ 0 & b_3 & a_5 & \cdots & & & \ddots\\ \vdots & \vdots &\vdots & \ddots \\ \hline & &&& a_2 & b_2 & 0&\cdots \\ &\ddots&&& b_2 & a_4 & b_4 & \cdots & & \\ &&\ddots&& 0 & b_4 & a_6 & \cdots & & \\ &&&& \vdots & \vdots &\vdots & \ddots \end{array}\right)$$ i.e. $T_n^{(2)}$ is the direct sum of two submatrices $\mathcal{O}_{n},$ $\mathcal{E}_n$. This converts the question from being about a single 2-tridiagonal matrix to a pair of standard 1-tridiagonal matrices. For instance, since $T_n^{(2)}=\mathcal{O}_{n}\oplus\mathcal{E}_n$ we have $\exp T_n^{(2)}=(\exp \mathcal{O}_{n})\oplus (\exp \mathcal{E}_{n}).$ There's an extensive amount of literature on such tridagonal matrices, so I won't attempt to cover it here.