I want to find the expected value of the $k$-th order statistic of a sample with size $n$, drawn from a Poisson distribution with mean $\lambda$. More precisely, I want to find $E(X_{(k)})$ where $X_i$'s are sampled from a Poisson distribution with probability $P(X=k) = \frac{\lambda^k e^\lambda}{k!}$, and then sorted to find the $k$-th order statistic.
Is there any closed form interpretation of the expected value?
The solution to this problem can be used in finding an upper bound on the expected number of balls in a problem known as balls into bins problem which has a use case in chain hashing. It's worth mentioning that in that problem, the distribution of balls in each bin can be approximated by an independent Poisson distribution, so the answer to this question can be used there to approximate the expected number of balls in heavily loaded bins.
Herbert A. David, H. N. Nagaraja - Order statistics-(2003) page 23. $$F_k(x)=\begin{cases} 1-\sum_\limits{i=0}^{k-1}e^{- \lambda F(x)}\frac{[\lambda F(x)]^i}{i!} & x<b,\\ 1 & x\geqslant b. \end{cases}$$ Where $F(x)$ is cdf on $[a,b]$.