A Vandermonde like determinant with exponentials

Question

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

Answer 1

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 $$ A=\left[\sum_{k=i}^{i+n-1}s_k^{j-1}e^{-s_k}\prod_{l=i,l\neq k}^{i+n-1}(s_k-s_l)^{-1}\right]_{\,i,j=1}^{\,m},\tag{1} $$ and define $r=m+n-1$. Note that this answer also holds for $n<m$.

$A$ can be factorized similar to this posting using the $r \times m$ Vandermonde matrix $$ V_{r,m}=\left[ s_\rho^{\mu-1}\right]_{\rho,\mu=1}^{r,m},\tag{2} $$ the diagonal matrix $$ E_r = \mathrm{diag}\,[e^{-s_\rho}]_{\rho=1}^r\tag{3} $$ and the matrix of inverse derivatives $$ D_{m,r}^{(o)}=\left[ \frac{1}{P_{\mu,o}'(s_\rho)} \begin{cases} 1 & \mu \leq \rho < m+\mu \\ 0 & \text{else} \end{cases} \right]_{\,\mu,\rho=1}^{\,m,r},\tag{4} $$ of the polynomials $$ P_{l,o}(s)=\prod_{\rho=l}^{l+o-1}(s-s_\rho)\tag{5} $$ according to $$ A = D_{m,r}^{(m)} E_{r} \, V_{r,m}.\tag{6} $$ Note that $D$ is not diagonal in this case, e.g., for $m=3$, $n=3$, where $r=5$: $$ D_{3,5}^{(3)}= \left( \begin{array}{ccccc} \frac{1}{\left(s_1-s_2\right) \left(s_1-s_3\right)} & \frac{1}{\left(s_2-s_1\right) \left(s_2-s_3\right)} & \frac{1}{\left(s_3-s_1\right) \left(s_3-s_2\right)} & 0 & 0 \\ 0 & \frac{1}{\left(s_2-s_3\right) \left(s_2-s_4\right)} & \frac{1}{\left(s_3-s_2\right) \left(s_3-s_4\right)} & \frac{1}{\left(s_4-s_2\right) \left(s_4-s_3\right)} & 0 \\ 0 & 0 & \frac{1}{\left(s_3-s_4\right) \left(s_3-s_5\right)} & \frac{1}{\left(s_4-s_3\right) \left(s_4-s_5\right)} & \frac{1}{\left(s_5-s_3\right) \left(s_5-s_4\right)} \\ \end{array} \right).\tag{7} $$

As the matrices $D$ and $V$ are not square, the determinant of $A$ does not factorize in a simple way. However, one could try to enlarge both matrices to $r \times r$ by inserting appropriate matrix elements $O(\epsilon)$, factorize and let $\epsilon\to 0$. Alternatively, one can naturally enlarge $D$ and $V$ and define the $r \times r$ matrix $$ B=D_{r,r}^{(m)} E_{r} \, V_{r,r},\tag{8} $$ where $\det B$ factorizes. The matrix $A$ then is the upper left $m \times m$ submatrix of $B$.