The classical discrete laws near the mode: complete local expansions for the negative binomial and hypergeometric distributions

Abstract

We derive complete local asymptotic expansions near the mode for the negative binomial and hypergeometric laws, complementing the binomial expansion obtained in the companion papers. The coefficients are given in closed Bernoulli-polynomial form and retain the exact lattice displacement from the mean. For each law an entropy normalisation replaces the Bernoulli polynomials by reciprocal Bernoulli polynomials and makes the natural reflection symmetry visible. The first-order coefficient also recovers the exact mode rule: in the Katz cases the vertex gives the classical threshold exactly, while in the hypergeometric case its explicit discrepancy is always too small to change the selected integer mode.