Moments of the negative multinomial distribution

The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. General formulas for the falling factorial moments of the negative multinomial distribution have been obtained in the past by Mosimann (1963), and similarly for cumulants by Withers & Nadarajah (2014). However, to the best of our knowledge, no one has ever calculated general formulas for the moments (although the moment generating function is known, see, e.g., Chapter~36 of Johnson et al. (1997), it is unpractical). In this paper, we fill this gap by providing general formulas for the central and non-central moments of the negative multinomial distribution in terms of binomial coefficients and Stirling numbers of the second kind. We use the formulas to give explicit expressions for all central moments up to the $4^{\text{th}}$ order and all non-central moments up to the $8^{\text{th}}$ order.