Gould series distributions with applications to fluctuations of sums of random variables

A two-parameter class of discrete distributions, Gould series distributions, generated by expanding a suitable parametric function into a series of Gould polynomials is discussed. A Gould series distribution occurs in fluctuations of sums of interchangeable random variables and particularly as the distribution of (i) the duration of the game in the theory of games of chance, (ii) the busy period in queueing processes and (iii) the time of emptiness in dam and storage processes. The probability generating function and the factorial moments of the Gould series distributions are obtained in close forms. It is pointed out that the name of the generalized general binomial (binomial or negative binomial) distribution of Consul and Jain is justified by the form of its generating function. Finally it is shown that the generalized general binomial distribution, under certain mild conditions, is the only member of the Gould series distributions which is closed under certain mild conditions, is the only member of the Gould series distributions which is closed under convolution