Irregular hypergeometric D-modules

Abstract

We study the irregularity of hypergeometric D-modules MA(β) via the explicit construction of Gevrey series solutions along coordinate subspaces in X=Cn. As a consequence, we prove that along coordinate hyperplanes the combinatorial characterization of the slopes of MA(β) given by M. Schulze and U. Walther (2008) in [23] still holds for any full rank integer matrix A. We also provide a lower bound for the dimensions of the spaces of Gevrey solutions along coordinate subspaces in terms of volumes of polytopes and prove the equality for very generic parameters. Holomorphic solutions of MA(β) at nonsingular points can be understood as Gevrey solutions of order one along X at generic points and so they are included as a particular case.