Entropy computations for discrete distributions: towards analytic information theory

We investigate the basic question of information theory, namely, evaluation of Shannon entropy, and a more general Renyi entropy, for some discrete distributions (e.g., binomial, negative binomial, etc.). We aim at establishing analytic methods (i.e., those in which complex analysis plays a pivotal role) for such computations which often yield estimates of unparalleled precision. The main analytic tool used is that of analytic poissonization and depoissonization. We illustrate our approach on the entropy evaluation of the binomial distribution