I have found the formula for the Probability Generating Function of negative multinomial distribution in Definition 8.1 of this chapter (https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat01250). It is as follows. $$G_x(\boldsymbol{t}) = \left(\mathcal{Q} - \sum^k_{i=1} P_{it_i}\right)^{-n}$$, where $n > 0$, $P_i > 0(i = i, \ldots, k)$, and $\mathcal{Q} - \sum^k_{i=1} P_{i} = 1$. I am still confused with the following:
- What do $\mathcal{Q}$ and $P_i$ mean?
- How can I use this to compute the coefficient of the terms in the series expansion of $\left(x_1, x_2, \ldots, x_m \right)^{-n}$, where, $m > 0$?
I am using this space to post the content of Section 2.4 which builds up to the equation I have used in the question.