Logarithmic A-hypergeometric series

TL;DR

This paper introduces a novel perturbation method for constructing logarithmic solutions of regular A-hypergeometric systems, expanding the techniques beyond classical and existing hypergeometric approaches.

Contribution

It proposes a new perturbation approach for logarithmic solutions in A-hypergeometric systems, differing from previous parameter perturbation methods.

Findings

Developed a new perturbation technique for hypergeometric series

Extended the construction of logarithmic solutions beyond classical methods

Provided theoretical framework for the new perturbation approach

Abstract

The method of Frobenius is a standard technique to construct series solutions of an ordinary linear differential equation around a regular singular point. In the classical case, when the roots of the indicial polynomial are separated by an integer, logarithmic solutions can be constructed by means of perturbation of a root. The method for a regular A-hypergeometric system is a theme of the book by Saito, Sturmfels, and Takayama. Whereas they perturbed a parameter vector to obtain logarithmic A-hypergeometric series solutions, we adopt a different perturbation in this paper.