Multinomial distributions in Shintani zeta class

Multidimensional stochastic models in mathematical finance and so on are now well-studied. As to obtain more properties of them, we focus on some multidimensional discrete distributions in relation to a class of multiple zeta functions. The class called “multidimensional Shintani zeta functions” was first introduced in Aoyama and Nakamura (Tokyo J Math 36:521–538, 2013), where a class of probability distributions called “multidimensional Shintani zeta distributions associated with the zeta functions is definable. In this paper, we show that this class includes many kinds of multidimensional discrete distributions. We pick up some cases of multidimensional Shintani zeta functions and introduce some classes of distributions which contain multinomial and negative multinomial distributions as their generalizations. More precisely, we give some necessary and sufficient conditions for the functions to generate probability distributions in view of zeta functions and consider their infinite divisibilities as well.