I was wondering if there is a special name for the following kind of tridiagonal matrices ? And if yes, are there any books or articles which talk about their properties ?
\begin{pmatrix} \alpha_1 & \beta_1 & 0 & 0 \\ \beta_{n-1} & \alpha_2 & \ddots & 0 \\ 0 & \ddots & \ddots & \beta_{n-1} \\ 0 & 0 & \beta_1 & \alpha_n \end{pmatrix}
(basically the lower diagonal is reversed compared to the upper diagonal)
I was not able to find any specific literature on it, but the closest terminology I would have to describe what you proposed would be a "centrosymmetric tridiagonal matrix".
In page of the following document (p.5) you can find a similar matrix in Eq. (2.5) to what you are looking for: https://core.ac.uk/download/pdf/81151355.pdf