Determinant of a specific, symmetric Toeplitz matrix

[Rlwe]

TL;DR Summary

Calculation of a ratio of determinants for a family of specific, symmetric Toeplitz matrices

Let us define matrix $\mathbf{B}_n=[b_{ij}]_{n\times n}$ as follows $$[b_{ij}]_{n\times n}:=\begin{cases} b_{ij} = \alpha\,,\quad j=i\\ b_{ij}=\beta\,,\quad j=i\pm1\\ b_{ij}=1\,,\quad \text{else}\end{cases}\,,$$ where $\alpha\,,\beta\in\mathbb{R}$ and $n\geq2$. $\mathbf{B}_4$, for example, looks like this $$\mathbf{B}_4=\begin{bmatrix} \alpha & \beta & 1 & 1 \\ \beta & \alpha & \beta & 1\\ 1 & \beta & \alpha &\beta \\ 1 & 1& \beta& \alpha \end{bmatrix}\,.$$ Is there any simple, closed form expression for a ratio of determinants $\frac{\det(\mathbf{B}_n)}{\det(\mathbf{B}_{n-1})}$?

 

[Maarten Havinga]

Not a simple one: I think you can only write out the determinant with all summands and factors and make that your formula. I doubt anything cancels out (tried a few low dimensional examples).