I'm interested in the following symmetric functions $s_k: \mathbb{R}_+^{k}\mapsto\mathbb{R_+}$: \begin{align*} s_k(x_1,x_2,\dots, x_k)&= \int\limits_{0=t_0<t_1<\dots<t_{k-1}<t_k=1} e^{-(t_1-t_0) x_1} e^{-(t_2-t_1)x_2}\dots e^{-(t_k-t_{k-1})x_k}dt_1 \dots dt_{k-1} \\ &=\sum_{i=1}^k \frac{e^{-x_i}} {\displaystyle\prod_{\substack{j=1,\dots,n\\ j\ne i}}(x_j-x_i)} \end{align*}

The formula on the second line holds whenever the $x_i$ are all distinct; however it becomes degenerate if $x_i=x_j$ for some $i\ne j$. Question: is there some "nice" pattern for the closed form expressions one obtains when some of the $x_i$ coincide? (Examples below.)

[ One motivation: consider a continuous-time Markov chain, with state space $\{1,2,3,\dots\}$, with rate $q_{ij}$ of jumping from state $i$ to state $j$ for $i\ne j$, and total rate $\lambda_i=\sum_{j\ne i} q_{ij}$ of leaving state $i$.

Suppose the chain starts in state $1$ at time $0$. Consider the event that during the time-interval $[0,1]$, the chain follows precisely the path $1\to2\to\dots\to (k-1)\to k$ and then stays in state $k$ until the end of the interval (making $k-1$ jumps in all).

This event has probability $q_{12}\dots q_{(k-1)k}s_k(\lambda_1, \dots, \lambda_k)$. ]

Anyway: everything is nice and smooth and closed-form expressions for cases with repeated entries can straightforwardly be derived (using, say, L'Hôpital or whatever) - however the details appear to get messy quite quickly. I'm wondering if there is some convenient or systematic way to write those expressions (other than just calculating each case).

For example $s_2(x, y)=(e^{-x}-e^{-y})/(y-x)$, and then $s_2(x, x)=e^{-x}$.

Taking e.g. $k=4$, we get (I think) \begin{align*} s_4(x,x,y,y)&=\frac{2(e^{-x}-e^{-y})}{(x-y)^3}+\frac{(e^{-x}+e^{-y})}{(x-y)^2} \\ s_4(x,x,x,y)&=\frac{e^{-y}-e^{-x}}{(x-y)^3}-\frac{e^{-y}}{(x-y)^2}+ \frac{e^{-y}}{2(x-y)} \\ s_4(x,y,z,z)&=\frac{e^{-x}}{(z-x)^2(y-x)} + \frac{e^{-y}}{(z-y)^2(x-y)} \\ &\,\,\,\,\,\,\,+e^{-z}\left\{\frac{(z-x)+(z-y)+(z-x)(z-y)+z^2}{(z-x)^2(z-y)^2}\right\}. \end{align*}

I'm aware that these things can be written in terms of multiple integrals with gamma densities; for example, if $k=k_1+\dots+k_m$ and there are $k_i$ arguments $x_i$ for each $i$, then I think \begin{equation*} s_k(\dots)=\int\limits_{0=t_0<t_1<\dots<t_{m-1}<t_m=1} \prod_{i=1}^m \left\{\frac{(t_i-t_{i-1})^{k_i-1}}{(k_i-1)!} e^{-(t_i-t_{i-1})x_i}\right\}dt_1\dots dt_{m-1}, \end{equation*} but I'm not sure that leads anywhere particularly nice.

Perhaps (beguiled by the beautiful form in the case of all $x_i$ distinct) I'm looking for something unreasonable, and these things just are what they are. Then again maybe someone has something nice to point out!