TITLE: How to calculate one Cauchy type determinant
QUESTION [7 upvotes]: As we know, a Cauchy determinant of size n admits the following explicit formula:
$$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y _i)}{\prod _{i,j=1}^n (x _i+y _j)}.$$
Is there something known about the following generalized Cauchy determinant?
$$\det \left(\frac{A_i+B_j}{x _i+y _j}\right) _{1\le i,j \le n}.$$
More specially, how does it go for 
$$\det \left(\frac{A_i+A_j}{x _i+x _j}\right) _{1\le i,j \le n}.$$
A simple case is for $x_i=i$. 
I wonder if there are some references about them. Thank you.

REPLY [2 votes]: A determinant of this type is discussed in these two publications on the two-dimensional square lattice Ising model on the rectangle:
Hucht, Alfred, The square lattice Ising model on the rectangle. I: Finite systems, J. Phys. A, Math. Theor. 50, No. 6, Article ID 065201, 23 p. (2017). ZBL1357.81154.
https://arxiv.org/abs/1609.01963 
Hucht, Alfred, The square lattice Ising model on the rectangle. II: Finite-size scaling limit, J. Phys. A, Math. Theor. 50, No. 26, Article ID 265205, 23 p. (2017). ZBL1369.82005.
https://arxiv.org/abs/1701.08722