  The hypergeometric distribution is a popular distribution, whose properties
have been extensively investigated. Generating functions of this distribution,
such as the probability-generating function, the moment-generating function,
and the characteristic function, are known to involve the Gauss hypergeometric
function. We elucidate that the existing formula of generating functions is
incomplete and provide the corrected formula. In view of the utility and
applicability of the hypergeometric distribution, the exact and correct
foundation is crucially important.
