Numerical analytical continuation of multivariate hypergeometric functions

This paper presents a general framework for the high-precision numerical evaluation and analytic continuation of multivariate hypergeometric functions by adapting methods from multi-loop Feynman integrals to construct Pfaffian systems via Laporta reduction and employing a Frobenius-based scheme to systematically track solutions across different Riemann sheets.

The Gist

Imagine you are trying to navigate a vast, foggy archipelago. This archipelago represents the world of multivariate hypergeometric functions. These are complex mathematical objects that appear everywhere in physics (like calculating particle collisions) and pure math.

The problem is that these functions are multivalued. Think of them like a spiral staircase that never ends. If you start at the bottom and walk in a circle, you don't end up on the same step; you end up on a different "floor" or "Riemann sheet" of the same building. If you take a different path around a pillar (a singularity), you might end up on a completely different floor.

For a long time, calculating the exact value of these functions at a specific point was like trying to guess which floor you were on without a map. Different computer programs would give you different answers for the same input because they were standing on different floors of the spiral staircase, and no one had a universal rulebook for how to switch between them.

This paper presents a new, high-precision GPS and navigation system for this archipelago. Here is how the authors built it, using simple analogies:

1. The Map: Turning Chaos into a Grid

First, the authors needed a way to describe the terrain. These functions are defined by infinite series (adding up endless numbers), which is hard to compute directly once you move away from the starting point.

• The Old Way: Trying to sum the infinite series directly.
• The New Way (Laporta Reduction): The authors treat the derivatives of these functions like a massive family of Feynman integrals (a concept from particle physics). They use a clever sorting algorithm (the Laporta algorithm) to realize that even though there are infinite derivatives, they can all be expressed in terms of a tiny, finite "master set" of derivatives.
• The Analogy: Imagine you have a library with infinite books. Instead of reading every single one, you realize that every book is just a remix of 5 specific "Master Books." The authors found these 5 Master Books and created a Pfaffian system—a set of rules that tells you exactly how to move from one derivative to another, like a strict set of traffic laws for the function.

2. The Vehicle: The Generalized Frobenius Method

Now that they have the rules (the map), they need a vehicle to travel along them. They use a method called the Frobenius method, but they upgraded it.

• The Problem: You can't drive a car in a straight line forever because the road might have potholes (singularities) or cliffs.
• The Solution: The authors don't try to drive the whole distance at once. Instead, they build a chain of overlapping safety bubbles (disks).
  • Inside the first bubble (near the start), they calculate the function's value with extreme precision.
  • They then drive to the edge of that bubble, where it overlaps with the next bubble.
  • They use the overlap to "glue" the two calculations together, effectively handing off the navigation to the next bubble.
• The Result: They can travel from the starting point to any destination in the complex plane, hopping from bubble to bubble, without ever falling off the edge.

3. The Compass: Tracking the "Floors" (Monodromy)

This is the most critical part. Because the functions are multivalued (like the spiral staircase), you need to know exactly which "floor" you are on.

• The Challenge: If you walk around a pillar (a singularity), you might end up on a different floor. How do you know which one?
• The Solution: The authors calculated Monodromy Matrices. Think of these as elevator buttons.
  • If you walk around a specific singularity, the Monodromy Matrix tells you exactly how the function changes. It's like a rule that says, "If you circle this pillar once, you go up 3 floors."
  • By combining their "bubble-hopping" travel with these "elevator buttons," they can systematically access any floor of the spiral staircase. They can prove that the answer Mathematica gives is the same as the answer Maple gives, just on a different floor, and they can translate between them.

4. The Road Rules: Branch Cuts

To make sure everyone agrees on what "Floor 1" means, you need to draw lines on the map where you aren't allowed to cross (Branch Cuts).

• The authors created a Canonical Path system. They defined a specific, step-by-step way to travel from the origin to any point (e.g., "First move along the real axis, then the imaginary axis").
• By following these strict road rules, they ensure that everyone using their tool starts on the same "principal branch" (the main floor), making the results consistent and reproducible.

Summary of What They Did

The authors created a software package (called HAPC) that:

1. Reduces complex, infinite mathematical problems into a manageable, finite set of rules.
2. Travels across the complex plane using a chain of overlapping calculation zones.
3. Tracks exactly which "version" (Riemann sheet) of the function you are on, allowing you to switch between them intentionally.
4. Delivers high-precision numbers for these functions, even in regions where they were previously impossible to calculate reliably.

They tested this on examples from particle physics (like Feynman diagrams) and showed that their method can reproduce results from other major software packages, but with the added superpower of knowing exactly how to switch between the different "floors" of the mathematical building.

In short: They built a universal, high-precision GPS for a multi-dimensional, multi-floor mathematical maze, complete with a rulebook for how to change floors without getting lost.

Technical Summary

Technical Summary: Numerical Analytical Continuation of Multivariate Hypergeometric Functions

Problem Statement
Multivariate hypergeometric functions constitute a rich class of special functions appearing in algebraic geometry, number theory, and high-energy physics, particularly in the computation of multi-loop Feynman integrals. These functions are often defined as solutions to holonomic systems of linear partial differential equations (PDEs). A central challenge in their application is the high-precision numerical evaluation of these functions outside their domain of absolute convergence. This requires robust methods for analytic continuation that can handle the multivalued nature of these functions, navigate complex singular loci, and consistently track changes across different Riemann sheets. Existing literature typically relies on connection formulas for specific subfamilies or integral representations (e.g., Mellin–Barnes contours), which often lack a unified, systematic framework for arbitrary multivariate cases.

Methodology
The authors propose a general framework that adapts techniques from multi-loop Feynman integral computations to the setting of multivariate hypergeometric functions. The methodology proceeds in three main stages:

1. Derivation of Pfaffian Systems via Laporta-style Reduction:
   Starting from a series representation of a multivariate hypergeometric function, the authors derive the associated holonomic system of linear PDEs with rational coefficients. They treat the infinite family of derivatives of the function as analogues to Feynman integrals. By applying a Laporta-style reduction algorithm to these differential relations, they express all derivatives in terms of a finite basis of "master derivatives." This process transforms the original higher-order PDE system into a first-order system in Pfaffian form ($d\mathbf{J} = \sum M_i d x_i \mathbf{J}$), where $\mathbf{J}$ is the vector of master functions.

2. Generalized Frobenius Method for Local Solutions:
   To solve the Pfaffian system, the authors employ a generalized Frobenius method. They restrict the system to a one-dimensional path in the complex space of variables. Around regular singular points, they construct local fundamental matrix solutions as power series (potentially including logarithmic terms for resonant cases). These local solutions are computed with arbitrary precision.

3. Analytic Continuation and Monodromy Tracking:
   The global solution is constructed by chaining overlapping disks along a prescribed path from a known boundary point (typically the origin) to the target evaluation point. The solution is propagated from one disk to the next by matching the truncated series in the overlap regions. Crucially, the framework explicitly manages the multivaluedness of the functions. By defining branch cuts and constructing specific "canonical paths" that avoid these cuts, the method ensures consistent evaluation on a chosen Riemann sheet. Furthermore, the authors demonstrate how to compute monodromy matrices by encircling singularities, allowing for the systematic enumeration of all possible values (branches) of the function at a given point.

Key Contributions

• Unified Framework: The paper establishes a general algorithm applicable to arbitrary multivariate hypergeometric functions defined by holonomic systems, rather than being limited to specific families like Appell or Lauricella functions.
• Laporta Adaptation: It successfully adapts the Laporta reduction algorithm, originally designed for Feynman integrals, to the reduction of differential operators for hypergeometric functions, enabling the construction of minimal Pfaffian systems.
• Controlled Analytic Continuation: The authors provide a systematic procedure for analytic continuation that explicitly tracks the Riemann sheet structure. This includes the construction of monodromy matrices and the definition of canonical paths to fix principal branches.
• HAPC Package: The methodology is implemented in a Mathematica package named HAPC (Hypergeometric Analytic Pfaffian Continuation), which serves as a proof-of-concept tool for high-precision numerical evaluation and $\epsilon$-expansions.

Results
The authors validate their framework through several examples:

• Elementary and Bivariate Cases: They demonstrate the recovery of multiple branches for the hypergeometric function $_2F_1$ and the Appell $F_3$ function, showing how different values returned by computer algebra systems (e.g., Mathematica vs. Maple) correspond to different Riemann sheets connected by monodromy transformations.
• Feynman Integrals: The method is applied to multi-loop Feynman integrals, specifically a massive two-loop sunset diagram and a non-planar two-loop vertex diagram. The authors successfully compute $\epsilon$-expansions for these integrals at kinematic points outside the convergence domain of the defining series, matching results obtained via multiple polylogarithms where available.
• Precision: The approach achieves high-precision numerical results (e.g., 30 decimal places) and correctly handles complex arguments and parameters dependent on the dimensional regularization parameter $\epsilon$.

Significance and Claims
The paper claims to fill a gap in the literature by providing a unified, systematically extensible framework for the numerical evaluation and analytic continuation of multivariate hypergeometric functions, a task that previously required ad hoc treatments for individual functions. The authors emphasize that their approach is particularly valuable for high-energy physics applications where parameters often depend linearly on $\epsilon$, necessitating precise Laurent expansions.

The authors are modest regarding the current scope of their implementation. They explicitly state that the current framework assumes regular singularities. While the Laporta reduction step applies to confluent cases (irregular singularities), the Frobenius-based continuation and monodromy analysis are not yet extended to handle the exponential asymptotic behavior and Stokes phenomena associated with irregular singularities. They identify the extension to confluent hypergeometric functions as a primary direction for future work. Similarly, the HAPC package is presented as a beta version and a proof of concept, with plans for performance optimization and expanded branch-selection controls in future iterations.