Recently I am trying to compute some variation of Cauchy determinant like this $$\det\left(\frac{1}{(x_i+y_j)^k}\right)$$ I have just learned from Google that Borchardt computed the above determinant when $k=2$.$$\det\left(\frac{1}{(x_i+y_j)^2}\right)=\det\left(\frac{1}{x_i+y_j}\right)\text{Perm}\left(\frac{1}{x_i+y_j}\right)$$

where $\text{Perm}(A)$ denotes the permanent of matrix $A$, here is the definition: definition of permanent Please recommend some material about this variation of Cauchy determinant to me. thanks very much.

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I cannot remember whether "the art of computer programming" has these information. – eccstartup Jun 25 '13 at 4:29
Yes, TAOCP has some problems in this area. Regards – Amzoti Jun 25 '13 at 4:30

For the case $k=2$ (and $k=1$) see also the article "Advanced Determinant Calculus" of Christian Krattenthaler: arxiv.org/abs/math/9902004. In $(3.9)$ and $(3.10)$, with $q=1$, formulas for the determinant of your matrix with $k=2$ (and $y'_j=-y_j$) is given. Here $(3.9)$ is due to Borchardt, and $(3.10)$ is a $(q)$-deformation of it.
For $k\ge 3$ there is a conjectured formula here: http://ipg.epfl.ch/~leveque/Conjectures/cauchy.pdf