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A Vandermonde like determinant with exponentials

After slight renumbering $(m,n)\mapsto (m,n)-1$ and horizontal reflection $j\mapsto m+1-j$, let's call the $m\times m$ matrix in the OP's determinant $$ A=\left[\sum_{k=i}^{i+n-1}s_k^{j-1}e^{-s_k}\...
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