Expressions for the Entropy of Binomial-Type Distributions

We develop a general method for computing logarithmic and log-gamma expectations of distributions. As a result, we derive series expansions and integral representations of the entropy for several fundamental distributions, including the Poisson, binomial, beta-binomial, negative binomial, and hypergeometric distributions. Our results also establish connections between the entropy functions and to the Riemann zeta function and its generalizations.