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Irregular Modified $A$-Hypergeometric Systems

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.