Complete corrected formula for generating functions of the hypergeometric distribution

TL;DR

This paper corrects and completes the formulas for the generating functions of the hypergeometric distribution, which are essential for accurate statistical analysis and applications involving this distribution.

Contribution

It provides the corrected and complete formulas for the generating functions involving the Gauss hypergeometric function.

Findings

Existing formulas are incomplete.

Corrected formulas involve the Gauss hypergeometric function.

Ensures accurate application of hypergeometric distribution properties.

Abstract

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.