Consider the polynomial $P(X) =\prod_{i=1}^k (X-x_i)^s$ and let $M$ be the corresponding confluent Vandermonde matrix. Concretely, here is what I mean by that. Define $$ M^{(0)} = \begin{pmatrix} 1 & 1 & \cdots & 1\\ x_1 & x_2 & \cdots &x_k\\ x^2_1 & x^2_2 & \cdots &x^2_k\\ \vdots\\ x^{ks-1}_1 & x^{ks-1}_2 & \cdots &x^{ks-1}_k \end{pmatrix} $$ which is the $ks\times k$ Vandermonde matrix. Let $D=\sum_i \partial/\partial x_i$ be an operator. Define $$ M^{(n)} = \frac{1}{n!}D^n M^{(0)} $$ Then $M=(M^{(0)}, M^{(1)}, \cdots, M^{(s-1)})$ which is $ks\times ks$. With that definition, now here is my question:

Is there a clean formula for the solution $F$ of the system $MF=B$? Equivalently, is $M^{-1}$ known in terms of, say, symmetric functions? For all intents and purposes we can define $M^{(n)}=D^n M^{(0)}$ instead (as confluent Vandermonde is sometimes defined) if it helps.