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Rlwe

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- 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})}##?