Incomplete A-Hypergeometric Systems

The incomplete beta function is an important special function in statistics. In modern theory of hypergeometric functions, we regard hypergeometric functions as pairings of twisted cycles and twisted cocycles. However, the incomplete beta function cannot be understood in this scheme; in other words, the domain of the integration is not cycle (incomplete). We will generalize the theory of A-hypergeometric systems for incomplete functions. We give a general study as well as a detailed study on an incomplete Gauss hypergeometric function.