Confluent Hermitian Cauchy matrices are diagonally signed (Q753900)

Given distinct complex numbers \(e_ 1,...,e_ t\) such that \(e_{\alpha}+\overline{e_{\beta}}\neq 0\) (1\(\leq \alpha,\beta \leq t)\), and given positive integers \(n_ 1,...,n_ t\), define the confluent Hermitian Cauchy matrix by \(C(e_ 1,n_ 1;e_ 2,n_ 2;...;e_ t,n_ t)=[C^{(\alpha,\beta)}]^ t_{\alpha,\beta =1},\) where \(C^{(\alpha,\beta)}\) are \(n_{\alpha}\times n_{\beta}\) Hankel matrices, which in turn are defined elementwise by \(C_{ij}^{(\alpha,\beta)}=\left( \begin{matrix} i+j-2\\ i-1\end{matrix} \right)(e_{\alpha}+\overline{e_{\beta}})^{1-i-j};\quad i=1,...,n_{\alpha},\quad j=1,...,n_{\beta}.\) A Hermitian matrix \(H=[H_{ij}]^ n_{i,j=1}\) with complex numbers \(H_{ij}\) is called diagonally signed if it has the property \(sign \det H=sign\prod^{n}_{i=1}H_{ii}\) both for H itself, and for all its principal submatrices. (Convention: sign x\(=\pm 1\) if \(\pm x>0\), sign 0\(=0).\) It is proved that the confluent Hermitian Cauchy matrix \(C(e_ 1,n_ 1;e_ 2,n_ 2;...;e_ t,n_ t)\) is diagonally signed.