# On transformations of A-hypergeometric functions

## TL;DR

This paper systematically studies transformations of A-hypergeometric functions using automorphisms of toric rings and integral representations, revealing that all linear transformations stem from polytope symmetries and applying this to analyze the Appell function F4.

## Contribution

It introduces a unified approach to understanding A-hypergeometric transformations via toric automorphisms and polytope symmetries, and applies it to specific functions like F4.

## Key findings

- All linear A-hypergeometric transformations originate from polytope symmetries.
- The approach links integral representations to automorphisms of toric rings.
- F4 does not have a certain Euler-type integral representation.

## Abstract

We propose a systematic study of transformations of $A$-hypergeometric functions. Our approach is to apply changes of variables corresponding to automorphisms of toric rings, to Euler-type integral representations of $A$-hypergeometric functions. We show that all linear $A$-hypergeometric transformations arise from symmetries of the corresponding polytope. As an application of the techniques developed here, we show that the Appell function $F_4$ does not admit a certain kind of Euler-type integral representation.