Inverse moments of univariate discrete distributions via the Poisson expansion

In this note we present a series expansion of inverse moments of a non-negative discrete random variate in terms of its factorial cumulants, based on the Poisson-Charlier expansion of a discrete distribution. We apply the general method to the positive binomial distribution and obtain a convergent series for its inverse moments with an error residual that is uniformly bounded on the entire interval 0<=p<=1.