  We study the solutions of irregular A-hypergeometric systems that are
constructed from Gr\"obner 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 complex n-space. 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.
