Logarithmic A-hypergeometric series II

TL;DR

This paper advances the construction of logarithmic series solutions for A-hypergeometric systems using a perturbing method, providing new fundamental systems and conditions for exponents, with applications to classical hypergeometric systems.

Contribution

It develops a perturbing method for constructing solutions and establishes conditions for fake exponents, extending the understanding of A-hypergeometric systems.

Findings

Constructed fundamental systems of solutions for A-hypergeometric systems.

Provided a sufficient condition for fake exponents to be actual exponents.

Applied results to Aomoto-Gel'fand and Lauricella systems.

Abstract

In this paper, following [6], we continue to develop the perturbing method of constructing logarithmic series solutions to a regular A-hypergeometric system. Fixing a fake exponent of an A-hypergeometric system, we consider some spaces of linear partial differential operators with constant coefficients. Comparing these spaces, we construct a fundamental system of series solutions with the given exponent by the perturbing method. In addition, we give a sufficient condition for a given fake exponent to be an exponent. As important examples of the main results, we give fundamental systems of series solutions to Aomoto-Gel'fand systems and to Lauricella's FC systems with special parameter vectors, respectively.