Which negative multinomial distributions are infinitely divisible? (Q1431522)

A general class of negative multinomial distributions on \(\mathbb N_0^n\) is defined by its probability generating functions to be of the form \((A(a_1,\dots,a_n)/A(a_1z_1,\dots,a_nz_n))^\lambda\) for some \(\lambda>0\) and some function \( A(\mathbf z)=\sum_{T\subset\{1,\dots,n\}}a_T\prod_{i\in T}z_i\). The author derives necessary and sufficient conditions on the coefficients of \(A\) such that the function defined above is a probability generating function for any \(\lambda>0\) and therefore obtaining all infinitely divisible negative multinomial distributions on \(\mathbb N_0^n\). An interesting comparison to a different class of infinitely divisible negative multinomial distributions studied by \textit{R. C. Griffiths} and \textit{R. K. Milne} [J. Multivariate Anal. 22, 13--23 (1987; Zbl 0618.62060)] is also given.