Computing Moments of Discrete Distributions by Finite Calculus

Moments of discrete distributions have been derived several times by operator methods: summing operator series, applying differential operators, or reading recurrences off differential equations for the generating functions. The remaining approach, summation by parts (the integration by parts of finite calculus), has been used repeatedly, but never as a method in its own right; we develop it systematically. Together with the absorption identity for binomial coefficients, it computes the factorial moments of the classical discrete distributions in one or two lines each, and raw moments follow mechanically through Stirling numbers. It continues to work when the moments have no closed form uniform in the parameters, producing exact identities and fixed-parameter recurrences. Along the way we re-derive in a few lines a recurrence of Kemp for the factorial moments of Fisher’s noncentral hypergeometric distribution, showing that, except in the central case, it makes every factorial moment an elementary function of the mean. We also observe that the Conway–Maxwell–binomial distribution with ν=2 is a noncentral hypergeometric distribution, and we obtain a recurrence with a drifting center for the factorial moments about the mean of the Poisson distribution, mirroring the classical recurrence for the central moments of the Gaussian distribution.