Euler--Mellin integrals and A-hypergeometric functions

TL;DR

This paper introduces Euler--Mellin integrals, generalizing Mellin transforms and Euler integrals, and interprets them as A-hypergeometric functions, revealing their analytic properties and connections to coamoebas and Mellin--Barnes integrals.

Contribution

It provides a new class of integrals called Euler--Mellin integrals, with explicit meromorphic continuation and a novel interpretation as A-hypergeometric functions.

Findings

Euler--Mellin integrals generalize classical Mellin and Euler integrals.

Explicit meromorphic continuation of these integrals is achieved.

Connections to coamoebas and Mellin--Barnes integrals are established.

Abstract

We consider integrals that generalize both the Mellin transforms of rational functions of the form 1/f and the classical Euler integrals. The domains of integration of our so-called Euler--Mellin integrals are naturally related to the coamoeba of f, and the components of the complement of the closure of the coamoeba give rise to a family of these integrals. After performing an explicit meromorphic continuation of Euler--Mellin integrals, we interpret them as A-hypergeometric functions and discuss their linear independence and relation to Mellin--Barnes integrals.