Irregular Hodge filtration of some confluent hypergeometric systems

TL;DR

This paper computes the irregular Hodge filtration for certain hypergeometric D-modules, providing explicit formulas for their irregular Hodge numbers using advanced transformations and reductions.

Contribution

It introduces a method to determine the irregular Hodge filtration for purely irregular hypergeometric systems, linking GKZ-systems and Gauss-Manin systems.

Findings

Explicit formulas for irregular Hodge numbers of hypergeometric systems

Reduction techniques from GKZ-systems to hypergeometric D-modules

Comparison results via Fourier-Laplace and Radon transformations

Abstract

We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric $\mathcal{D}$-modules. We obtain in particular a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss-Manin systems of Laurent polynomials via Fourier-Laplace and Radon transformations.