Incomplete A-Hypergeometric Systems

The incomplete gamma function and incomplete beta function are important special functions in statistics. In modern theory of special functions, we regard hypergeometric functions as pairings of twisted cycles and twisted cocycles [1, Chap 2]. However, domains of integrations of these incomplete functions are not cycles. Solutions of A-hypergeometric systems ([3], [7, p.221]) have integral representations and domains of integrations are cycles. Some important integrals with parameters such as marginal likelihood functions in statistics, e.g., [4], look like A-hypergeometric, but domains of integrations are not cycles. Being motivated with these functions, we want to generalize the theory of A-hypergeometric systems to that for incomplete functions. We also hope that our theorems and formulas are useful for numerical and asymptotic evaluations of incomplete A-hypergeometric functions. This paper is a first step toward this direction. We give general discussions as well as detailed discussions on ∆1 × ∆1-incomplete hypergeometric functions. A definition of incomplete generalized hypergeometric functions is given by Chardhry and Qadir [2]. A study of relations of these two definitions is a future problem.