Let $n\geq m$ be non negative integers, and consider a list of $(n+m+1)$ distinct numbers (complex or real). I am interested in getting a closed form formula for the following determinant: $\det\left[\sum_{k=i}^{n+i}s_k^{m+1-j}e^{-s_k}\left[\prod_{l=i,l\neq k}^{n+i}(s_k-s_l)\right]^{-1}\right]_{i,j=1\ldots m+1}$, i.e.

$$ \det\left(\begin{array}[ccc] \,\sum_{k=1}^{n+1}s_k^me^{-s_k}\left[\prod_{l=1,l\neq k}^{n+1}(s_k-s_l)\right]^{-1}& \ldots &\sum_{k=1}^{n+1}e^{-s_k}\left[\prod_{l=1,l\neq k}^{n+1}(s_k-s_l)\right]^{-1} \\ \vdots & \ddots & \vdots \\ \sum_{k=m+1}^{n+m+1}s_k^me^{-s_k}\left[\prod_{l=m+1,l\neq k}^{n+m+1}(s_k-s_l)\right]^{-1}& \ldots &\sum_{k=m+1}^{n+m+1}e^{-s_k}\left[\prod_{l=m+1,l\neq k}^{n+m+1}(s_k-s_l)\right]^{-1} \end{array}\right) $$.

This determinant can be rewritten as the determinant of an alternant matrix $\left(F_{j}(S_i)\right)_{i,j=1\ldots m+1}$, for suitable multivariate functions $F_j:\mathbb{C}^{n+1}\rightarrow\mathbb{C}$.

If the exponential are not there, then this determinant should some king of Schur polynomial. I am wondering if this determinant can be interpreted in terms of representation theory.

A related post : A class of matrix determinants between Wronskians and Vandermondes