Moments of the multivariate hypergeometric distribution

Question

Assume the Wikipedia definitions for the multivariate hypergeometric distribution, i.e. let $(X_1,\dots X_c)$ have multivariate hypergeometric distribution with $c\in\mathbb{N}$, $(K_1,\dots,K_c)\in\mathbb{N}^c$, $N=\sum_{i=1}^c K_i$ and $n\in\{0,\dots,N\}$.

Question

Is there a formula for moments of form $\mathbb{E}[\prod_{i=1}^cX_i^{m_i}]$, where $m_1,\dots,m_c\in\mathbb{N}$?

Answer 1

There is an article On the Moments of Multivariate Discrete Distributions Using Finite Difference Operators, where an expression is dirived: $$\mathbb{E}[\prod_{i=1}^lX_i^{m_i}]={N\choose n}^{-1}\prod_{j=1}^{l}\sum_{i=1}^{m_i}{K_j\choose i}{N-i\choose n-i}\Delta^jO^{m_i}$$ for $l\le c$.

However, I think the proof is wrong.