HyperPrecision: A Mathematica package for High-Precision Numerical Evaluation of Multivariate Hypergeometric Functions

This paper introduces HyperPrecision, a Mathematica package that enables high-precision numerical evaluation of multivariate hypergeometric functions and their Laurent expansions by automatically constructing Pfaffian systems, reducing them to ordinary differential equations along a contour, and solving them via the Frobenius method to overcome convergence limitations in physics and mathematics applications.

The Gist

Imagine you are trying to navigate a vast, complex landscape using a map. In the world of advanced mathematics and physics, this landscape is filled with "Multivariate Hypergeometric Functions." These are incredibly powerful mathematical tools used to describe everything from the behavior of subatomic particles to the structure of the universe.

However, there's a catch: the standard maps (mathematical formulas) for these functions only work in a tiny, safe neighborhood called the "convergence region." If you try to use these formulas outside that neighborhood—where the real action often happens in physics—they break down, give wrong answers, or simply refuse to work. Getting from the safe zone to the dangerous, interesting zones usually requires a very difficult, manual process called "analytic continuation," which is like trying to rebuild a bridge while you are already walking across a chasm.

Enter HyperPrecision: The GPS for Mathematical Landscapes

The paper introduces HyperPrecision, a new software package (written for the computer program Mathematica) that acts like a high-tech GPS for these mathematical functions. Instead of relying on the broken local maps, HyperPrecision builds a new, robust route automatically.

Here is how it works, using a few simple analogies:

1. The Problem: The "Dead Zone"

Think of the defining series of these functions as a flashlight. It shines brightly and clearly only in a small circle (the convergence region). If you step outside that circle, the light goes out, and you are in the dark. Physicists need to know what the function looks like far outside that circle, but they can't just walk there because the "ground" (the math) is unstable.

2. The Solution: Building a "Tunnel" (The Pfaffian System)

HyperPrecision doesn't try to walk around the dark area. Instead, it builds a tunnel through it.

• The Blueprint: First, the software looks at the mathematical definition of the function and automatically figures out the "rules of the road" (a system of differential equations) that the function must follow everywhere, not just in the safe zone.
• The Tunnel: It then draws a straight line (a contour) from the starting point (where the math is easy and known) to the destination point (where the physicist needs the answer).
• The Journey: It treats this line as a one-way street and solves the equations step-by-step along this path. It starts with a known value at the beginning and "drives" the solution forward to the target.

3. The "Frobenius" Engine

To drive this tunnel, the package uses a method called the Frobenius method. Imagine you are walking along a path and taking small, precise steps. At each step, you check your position against the rules of the road to make sure you haven't drifted off course. HyperPrecision does this with extreme mathematical precision, ensuring that even if the path goes through "rough terrain" (singularities or complex numbers), it stays on track.

4. The "Laurent" Expansion (The Zoom Lens)

Often, physicists don't just want a single number; they want to know how the function behaves when a tiny parameter (called $\epsilon$) changes slightly. It's like looking at an object through a zoom lens to see the fine details.
HyperPrecision is smart enough to not just calculate one number, but to calculate a whole "zoomed-in" view (a Laurent expansion). It does this by taking many snapshots at slightly different settings and then stitching them together to create a smooth, high-definition picture of the function's behavior.

What Can It Do?

The paper demonstrates that HyperPrecision is a general-purpose tool. It isn't limited to just one type of function. It successfully handles:

• Appell Functions: Common in particle physics.
• Horn Series: A broad family of complex functions.
• Lauricella Functions: Used in multi-loop calculations.

The authors tested it against known mathematical identities and other software, and it matched perfectly, even in places where other tools failed or gave up.

Real-World Applications Mentioned

The paper shows the package being used in three specific areas of physics:

1. Angular Integrals: Calculating how particles scatter and interact in quantum field theory.
2. Cosmological Correlators: Understanding the patterns of the early universe (inflation) and how massive fields influenced the formation of structures.
3. Holographic Correlators: Studying the relationship between gravity and quantum mechanics in specific theoretical models (Dp-branes).

The Bottom Line

HyperPrecision is a new tool that automates the hardest part of working with these complex mathematical functions. It takes a function that is only defined in a small, safe area and automatically extends it to any point a physicist might need, with high precision and without requiring the user to manually perform difficult mathematical gymnastics. It turns a "dead end" in mathematical navigation into a smooth, drivable road.

Technical Summary

Technical Summary of "HyperPrecision: A Mathematica package for High-Precision Numerical Evaluation of Multivariate Hypergeometric Functions"

Problem Statement
Multivariate hypergeometric functions (MHFs) are ubiquitous in mathematics and physics, appearing in contexts ranging from quantum field theory (Feynman integrals, scattering amplitudes) to cosmology (correlators) and holography. However, their high-precision numerical evaluation remains a significant challenge. MHFs are typically defined by infinite series that converge only within restricted domains of the argument space. Evaluating these functions in physically relevant kinematic regions often requires analytic continuation, which is generally non-trivial and unavailable in closed form for arbitrary Horn-type functions. Furthermore, many physics applications require not just the function value, but its Laurent expansion in a small parameter $\epsilon$ (e.g., dimensional regularization), adding another layer of complexity. Existing automated tools are often restricted to specific classes of functions, specific numbers of variables, or pre-configured systems, lacking a general framework for arbitrary Horn-type MHFs.

Methodology
The paper introduces HyperPrecision, a Mathematica package designed to numerically evaluate general complete Horn-type multivariate hypergeometric functions and their $\epsilon$-expansions. The methodology relies on the differential-equation method, inspired by techniques used for Feynman integrals. The core algorithm proceeds in three main stages:

1. Dynamic Derivation of the Pfaffian System:
   Instead of relying on hard-coded differential equations, the package automatically constructs the system of partial differential equations (PDEs) satisfied by a given MHF. Starting from the series definition, it identifies the annihilating operators. Using the modular arithmetic and rational reconstruction capabilities of the external library FiniteFlow, the package reduces the system to a Pfaffian form ($dg = \Omega g$). This involves constructing a basis for the quotient of the rational Weyl algebra by the left ideal generated by the annihilating operators, effectively determining the holonomic rank and the connection matrices $\Omega_i$ without suffering from intermediate expression swell common in symbolic Gröbner-basis methods.

2. Numerical Transport via Frobenius Method:
   To evaluate the function at a target point (potentially outside the convergence region), the multivariate Pfaffian system is restricted to a one-dimensional straight-line contour connecting the origin (where the series converges and boundary conditions are known analytically) to the target point. This reduces the problem to an ordinary differential equation (ODE). The package employs the DESolver library (part of the AMFlow package) to solve this ODE numerically. It utilizes the Frobenius method, constructing generalized power-series ansätze around singular points and matching them in their overlapping regions of convergence to transport the solution from the origin to the target.

3. Laurent Expansion Reconstruction:
   For functions depending on a small parameter $\epsilon$, the package avoids symbolic expansion of the Pfaffian system. Instead, it evaluates the system on a finite grid of numerical $\epsilon$ values. The coefficients of the Laurent expansion are then reconstructed via interpolation using the EpsFit[] command. This approach allows the computationally expensive transport steps to be parallelized across different $\epsilon$ values.

Key Contributions

• Automation: The package dynamically derives the Pfaffian system for any input Horn-type MHF, removing the need for manual derivation and hard-coding of differential equations.
• Generality: It supports a wide class of functions, including Appell ($F_1$–$F_4$), Horn ($G$- and $H$-series), and Lauricella ($F_A, F_B, F_C, F_D$) functions, with arbitrary numbers of variables.
• Analytic Continuation: It handles evaluation at arbitrary target points, including those lying on singular curves, by automatically managing branch cuts (via the IDelta option) and performing analytic continuation along the transport contour.
• High-Precision $\epsilon$-Expansion: It provides a robust method for computing Laurent expansions in $\epsilon$ up to arbitrary orders with high numerical precision.

Results and Validation
The authors validate the package through extensive benchmarks:

• Cross-Validation: Results for Appell and Lauricella functions were compared against the PrecisionLauricella package, in-house C++ solvers, and other specialized Mathematica packages (AppellF1.wl, etc.). Agreement was found to the full precision (15+ digits) in all testable cases.
• Reduction and Connection Formulae: The package was tested against known reduction formulae (e.g., reducing to logarithms and polylogarithms) and connection formulae (e.g., Euler transformations, relations between Appell functions).
• Feynman Integrals: The package successfully evaluated one-loop bubble and two-loop sunset integrals, which map to Appell $F_4$ and Lauricella $F_C^{(3)}$ functions respectively. These results were cross-checked against the AMFlow package, showing perfect agreement.
• Singular Points: The package demonstrated the ability to yield correct numerical values at specific singular points (e.g., $x=y$ for Appell $F_1$, boundary points for $F_4$) through asymptotic expansions, though stability can vary.
• Performance: Evaluation times were shown to scale with the holonomic rank of the system. The package successfully handled functions with up to six variables.

Applications Demonstrated
The paper illustrates the utility of HyperPrecision in three distinct physics applications:

1. Angular Integrals: Evaluation of massless and double-massive angular integrals in perturbative QFT, achieving expansions to $O(\epsilon^{10})$ with 50-digit precision, surpassing existing analytic results.
2. Cosmological Correlators: Direct evaluation of the bispectrum shape function for double-massive exchange in the physical region, bypassing the need for the delicate manual analytic continuation required in previous literature.
3. Holographic Correlators: Evaluation of three-point scalar correlators in non-conformal Dp-brane backgrounds, where the physical region lies outside the convergence domain of the defining Appell $F_4$ series.

Significance and Limitations
The paper positions HyperPrecision as a versatile tool for the scientific community, filling a gap for the high-precision numerical evaluation of general Horn-type MHFs. Its significance lies in automating the complex analytic continuation process, allowing physicists to evaluate functions directly in physical kinematic regions without manual intervention.

The authors maintain a modest scope regarding current limitations:

• The package is currently restricted to complete Horn-type functions with real arguments; complex target points are not fully supported.
• Degenerate cases where the holonomic rank drops for special parameter values are not yet handled automatically.
• Performance is dependent on the holonomic rank and the complexity of the Pfaffian system.
• Future work is suggested to include native C++ integration for speed, handling of complex arguments, and the reconstruction of analytic expressions from high-precision numerical data.