Hypergeometric High Precision Evaluation

---
name: hypergeometric-high-precision-evaluation
description: Methodology for high-precision numerical evaluation of multivariate hypergeometric functions using Pfaffian systems and contour restriction. Applicable to quantum field theory, string theory, number theory, and statistics computations.
category: mathematics
arxiv_id: "2605.30216"
arxiv_url: https://arxiv.org/abs/2605.30216
date: 2026-05-29
trigger: hypergeometric, pfaffian, high-precision, horn-type, mathematica, multivariate, laurent expansion, quantum field theory, numerical evaluation
---

# High-Precision Hypergeometric Function Evaluation Methodology

## Background
Horn-type multivariate hypergeometric functions appear widely in quantum field theory, string theory, number theory, and statistics. Their defining series converge only in restricted domains, and analytic continuation beyond these domains is non-trivial.

## Core Methodology (from arXiv:2605.30216)

### Key Insight
Automatically construct the Pfaffian system of partial differential equations for a given hypergeometric function, then restrict it to a one-dimensional contour for efficient numerical evaluation.

### Pattern Steps

1. **Identify the Horn-type hypergeometric function** from the problem context
   - Recognize the series structure and parameters
   - Determine the convergence domain

2. **Construct the Pfaffian system**
   - Derive the system of PDEs satisfied by the function
   - Express as a matrix-valued first-order system: dF = A(x)F

3. **Contour restriction**
   - Choose a one-dimensional path from a known evaluation point to the target
   - Restrict the Pfaffian system along this contour
   - Reduces multivariate PDE to ODE system

4. **Numerical integration**
   - Use high-precision ODE solvers along the contour
   - Automatic Laurent expansion in small parameter ε
   - Achieve arbitrary precision evaluation

### Applications

- **Quantum Field Theory**: Feynman integral evaluation, dimensional regularization
- **String Theory**: Amplitude computations, period integrals
- **Number Theory**: Special values of L-functions, modular forms
- **Statistics**: Multivariate distribution functions

### Implementation Notes

- The HyperPrecision Mathematica package automates steps 2-4
- Key advantage: avoids series convergence limitations
- Works for general Horn-type functions, not just Appell/Gauss
- Laurent expansion support enables ε-expansion in dimensional regularization

### Pitfalls

- Contour choice affects numerical stability; avoid singular points
- Pfaffian system construction can be computationally expensive for high-rank functions
- High-precision arithmetic significantly slower than machine precision
- Need careful handling of branch cuts in analytic continuation

## Reusable Skill Pattern

**When to use**: Any computation requiring evaluation of multivariate hypergeometric functions outside their convergence domain, particularly in QFT Feynman integrals or special function computations.

**Input**: Hypergeometric function definition (series/parameters), target evaluation point, desired precision

**Output**: High-precision numerical value, with optional Laurent expansion coefficients

**Validation**: Compare with known special cases (Gauss 2F1, Appell F1) in their convergence domains