Asymptotic factorial powers expansions for binomial and negative binomial reciprocals

By considering the variance formula for a shifted reciprocal of a binomial proportion, the asymptotic expansions of any order for first negative moments of binomial and negative binomial distributions truncated at zero are obtained. The expansions are given in terms of the factorial powers of the number of trials n n . The obtained formulae are more accurate than those of Marciniak and Wesołowski (1999) and simpler, as they do not involve the Eulerian polynomials.