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Question. Is there a closed-form expression for the determinant of a $n \times n$ matrix $A$ with entries $$ A_{i,j} = \frac{1 - \delta_{i, j}}{z_i - z_j}, \qquad 1\leq i, j\leq n, $$ where $z_i$ is a sequence of pairwise distinct complex numbers. The values on the diagonal are $0$.

It looks like a limit of a Cauchy matrix, but with $0$ on the diagonal, I'm particularly interested in the case where the $z_i$ have unit norm.

I welcome any reference, suggestion or special case,

Thanks,

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  • $\begingroup$ Nice problem, but no closed formula. :-( $\endgroup$
    – T. Amdeberhan
    Commented Feb 15, 2017 at 13:13
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    $\begingroup$ Note that it is anti symmetric. In particular, for odd n the determinant will be 0 and for even n it will be the square of some rational function. $\endgroup$
    – Ofir Gorodetsky
    Commented Feb 15, 2017 at 13:19
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    $\begingroup$ The denominators factor nicely: $\prod_{i<j}(z_i-z_j)^2$. $\endgroup$
    – Martin Rubey
    Commented Feb 15, 2017 at 21:01
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    $\begingroup$ Looking at the Pfaffian as a symmetric function doesn't look completely hopeless. I get $6m_{1 1 1 1} - m_{2 1 1} + m_{2 2}$ and $90m_{2 2 2 2 2 2} - 12m_{3 2 2 2 2 1} + 4m_{3 3 2 2 1 1} + 3m_{3 3 2 2 2} - 3m_{3 3 3 2 1} + 6m_{3 3 3 3} + 3m_{4 2 2 2 1 1} - 6m_{4 2 2 2 2} - 3m_{4 3 2 1 1 1} + m_{4 3 2 2 1} + m_{4 3 3 1 1} - m_{4 3 3 2} + 6m_{4 4 1 1 1 1} - m_{4 4 2 1 1} + m_{4 4 2 2}$ where m are the monomial symmetric functions $\endgroup$
    – Martin Rubey
    Commented Feb 15, 2017 at 21:52
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    $\begingroup$ This matrix is sometimes known as Trummer's matrix -- this might be a useful search term to you, but my Google-fu returned nothing. $\endgroup$
    – Federico Poloni
    Commented Feb 16, 2017 at 19:46

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