The structure of bivariate rational hypergeometric functions

TL;DR

This paper characterizes all codimension-two lattice configurations that admit stable rational hypergeometric functions, linking their structure to toric residues and exploring the rationality of certain bivariate factorial quotient series.

Contribution

It provides a complete description of stable rational $A$-hypergeometric functions for specific lattice configurations, connecting them to toric residues.

Findings

All stable rational $A$-hypergeometric functions are described by toric residues.

The structure of these functions is characterized for codimension-two lattice configurations.

Applications include studying the rationality of bivariate series with factorial quotient coefficients.

Abstract

We describe the structure of all codimension-two lattice configurations $A$ which admit a stable rational $A$-hypergeometric function, that is a rational function $F$ all whose partial derivatives are non zero, and which is a solution of the $A$-hypergeometric system of partial differential equations defined by Gel'fand, Kapranov and Zelevinsky. We show, moreover, that all stable rational $A$-hypergeometric functions may be described by toric residues and apply our results to study the rationality of bivariate series whose coefficients are quotients of factorials of linear forms.