Given the bidiagonal matrix $$ \mathbf{A}=\begin{bmatrix} a_1 & b_1 & 0 & 0 & \dots & 0 & 0\\ 0 & a_2 & b_2 & 0 & \dots & 0 & 0 \\ 0 & 0 & a_3 & b_3 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & \dots & a_{n-1} & b_{n-1} \\ 0 & 0 & 0 & 0 & \dots & 0 & a_n \end{bmatrix} $$
is there any simplified formulas for the entries of $\mathbf{A}^k$ and $\exp(\mathbf{A})$ where $k$ is any positive integer. Please suggest books or articles if possible.