Summary: In a recent article [the second author et al., J. Stat. Phys. 146, No. 2, 264–280 (2012; Zbl 1238.81142)] generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal probability distributions was a key point to allow to give them a statistical interpretation in terms of probabilities. In this article we present an approach based on generating functions that solves the previous difficulties. Our main theorem makes explicit the conditions under which those formal probability distributions are always non-negative. Therefore, the constraints of non-negativeness are automatically fulfilled giving a complete characterization in terms of generating functions. A large number of analytical examples becomes available.
©2012 American Institute of Physics