Irregular Modified $A$-Hypergeometric Systems

TL;DR

This paper investigates the irregularity and asymptotic behavior of modified $A$-hypergeometric systems, introducing new tools to analyze their solutions and linking divergent series to holomorphic solutions.

Contribution

It adapts the umbrella concept to study irregularity in modified systems, providing new insights into slopes, Gevrey solutions, and integral representations.

Findings

Analysis of slopes and Gevrey series solutions

Laplace integral representations for divergent series

Connection between Gevrey asymptotics and holomorphic solutions

Abstract

A modified $A$-hypergeometric system is a system of differential equations for the function $f(t^w \cdot x)$ where $f(y)$ is a solution of an $A$-hypergeometric system in $n$ variables and $w$ is an $n$ dimensional integer vector, which is called the weight vector. We study the irregularity of modified systems by adapting to this case the notion of umbrella introduced by M. Schulze and U. Walther. Especially, we study slopes and Gevrey series solutions. We develop some applications of this study. Under some conditions we give Laplace integral representations of divergent series solutions of the modified system and we show that certain Gevrey series solutions of the original $A$-hypergeometric system along coordinate varieties are Gevrey asymptotic expansions of holomorphic solutions of the $A$-hypergeometric system.