Nilsson solutions for irregular A-hypergeometric systems

Abstract

We study the solutions of irregular A-hypergeometric systems that are constructed from Grobner degenerations with respect to generic positive weight ¨ vectors. These are formal logarithmic Puiseux series that belong to explicitly described Nilsson rings, and are therefore called (formal) Nilsson series. When the weight vector is a perturbation of (1, . . . , 1), these series converge and provide a basis for the (multivalued) holomorphic hypergeometric functions in a specific open subset of C n . Our results are more explicit when the parameters are generic or when the solutions studied are logarithm-free. We also give an alternative proof of a result of Schulze and Walther that inhomogeneous A-hypergeometric systems have irregular singularities.