MGF of the multivariate hypergeometric distribution Does the multivariate hypergeometric distribution, for sampling without replacement from multiple objects, have a known form for the moment generating function?
 A: Yes, it has, but it doesn't seem to be well known. It has the same form as the mgf of the (univariate) hypergeometric as given in https://en.wikipedia.org/wiki/Hypergeometric_distribution, so let us start with that. 
The hypergeometric probability mass function is
$$
   f(x) = \frac{\binom{a}{x}\binom{b}{n-x}}{\binom{a+b}{n}}
$$
and then the mgf can be written
$$
   M(t)=\frac{_2F_1(-n,-a;b-n+1;e^t)}{_2F_1(-n,-a;b-n+1;1)}
$$
and other forms can be given. Kendall's Advanced theory of Statistics gives it as the solution of a differential equation, while there is a short paper by Roanld Lessing "An Alternative Expression for the Hypergeometric Moment Generating Function" which gives it as an $n$th derivative of some related function. 
Generalization to the multivariate case is given in the paper by K G Janardan & G P Patil "A Unified Approach for a class of Multivariate Hypergeometric Models" in Sankhya. First some notation.
A finite population consists of $s+1$ subpopulation, indexed $0,1,\dotsc,s$. The population size is $N$ and $N_i$ the subpopulation sizes. We are sampling without replacement , sample size $n$. $X$ is the random vector $X=(X_1, X_2, \dotsc, X_s)$. The multivariate hypergeometric (MH) pmf can then be written as
$$
   mh(x;n,N)= \frac{\binom{N_0}{n-\sum x_i}\prod_1^s\binom{N_i}{x_i}}{\binom{N}{n}}
$$
and the multivariate moment generating function can then be written as
$$
 M_X(t_1,\dotsc,t_s)=\frac{N_0^{(n)}}{N^{(n)}} {}_2F_1(-n,-N_1,\dotsc,-N_s;N_0-n+1;e^{t_1},\dotsc,e^{t_s})
$$
and the symbol $\alpha^{(n)}=\alpha (\alpha-1) \dotsm (\alpha-n+1)$. 
This paper gives similar information also for related distributions like mv (multivariate) multinomial, mv inverse hypergeometric, mv negative hypergeometric, mv negative inverse hypergeometric and extensions like mv Polya and mv inverse Polya. 
