[Paper Review] Euler--Mellin integrals and A-hypergeometric functions

Christine Berkesch, Jens Forsgård+1 more|arXiv (Cornell University)|Mar 31, 2011

Polynomial and algebraic computation10 references4 citations

TL;DR

This paper introduces Euler–Mellin integrals as a generalization of Mellin transforms of rational functions, linking them to A-hypergeometric functions via meromorphic continuation over non-compact, simply connected domains. The key contribution is showing that these integrals yield entire A-hypergeometric functions in parameters and provide a full basis of solutions at generic parameters, with explicit examples revealing special solutions at rank-jumping parameters.

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.

Motivation & Objective

• To generalize Mellin transforms of rational functions by introducing Euler–Mellin integrals over non-compact, simply connected domains.
• To perform explicit meromorphic continuation of these integrals, removing convergence restrictions on parameters.
• To establish a direct link between Euler–Mellin integrals and A-hypergeometric functions, showing they satisfy the A-hypergeometric system with parameter β = -(t,s).
• To demonstrate that these integrals provide a full basis of solutions for A-hypergeometric systems under certain conditions on the matrix A.
• To analyze the behavior of solutions at non-generic parameters, particularly at rank-jumping parameters, using explicit expansions and coamoeba-based integration domains.

Proposed method

• Define Euler–Mellin integrals as multivalued integrals over non-compact, simply connected domains in (C*)^n, parameterized by s and t.
• Use coamoeba theory to identify connected components Θ of T^n ∓ cl(A'f), which define the domains of integration via Arg^(-1)(θ).
• Perform explicit meromorphic continuation of the integrals using Pochhammer symbol expansions and analytic continuation techniques, removing initial convergence restrictions.
• Show that the meromorphically continued integrals satisfy the A-hypergeometric system H_A(β) with β = -(t,s), using the A-hypergeometric framework from GKZ and Saito-Sekiguchi-Tagaki.
• Compare Euler–Mellin integrals to Mellin–Barnes integrals, proving at least as many linearly independent solutions exist via Theorem 5.6 and Corollary 5.7.
• Use explicit series expansions at special parameters (e.g., (s,t) = (-2,-1)) to detect additional solutions not captured by standard methods, revealing Laurent polynomial solutions at rank-jumping parameters.

Experimental results

Research questions

• RQ1 How can Mellin transforms of rational functions be generalized to non-compact domains while preserving analytic structure?
• RQ2 Can Euler–Mellin integrals be meromorphically continued to yield entire functions in the parameters, and what is the nature of their singularities?
• RQ3 To what extent do Euler–Mellin integrals span the solution space of A-hypergeometric systems, especially at non-generic parameters?
• RQ4 How do Euler–Mellin integrals relate to Mellin–Barnes integrals in terms of linear independence and solution basis?
• RQ5 What role do coamoebas play in defining the integration domains and capturing special solutions at rank-jumping parameters?

Key findings

• The meromorphic continuation of Euler–Mellin integrals results in a function that is entire in the parameters (s,t), with singularities confined to specific hyperplanes.
• For generic β, Euler–Mellin integrals provide a full basis of solutions to the A-hypergeometric system H_A(β), as shown in Proposition 4.3.
• At the rank-jumping parameter β = (1,2) for A = [1 1 1 1; 0 1 3 4], the Euler–Mellin integral yields two additional entire functions Φ₁ and Φ₂ that span the extra solution space, with Φ₁(-2,-1,c) = 2c₂²/c₁ and Φ₂(-2,-1,c) = 2c₃²/c₄.
• The expansion of the integral at (s,t) = (-2,-1) reveals a decomposition involving (4t-s+2), (s+2), and their product, explaining the vanishing of the main integral and the emergence of special solutions.
• Euler–Mellin integrals are shown to be at least as linearly independent as Mellin–Barnes integrals, with Corollary 5.7 confirming linear independence at generic β.
• The method captures Laurent polynomial solutions at non-generic parameters, providing a direct computational link between local cohomology and solution space structure.