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arxiv: 2605.30216 · v1 · pith:3AR6FROPnew · submitted 2026-05-28 · ✦ hep-ph · hep-th· math-ph· math.MP

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

Sumit Banik , Souvik Bera This is my paper

Pith reviewed 2026-06-29 06:28 UTC · model grok-4.3

classification ✦ hep-ph hep-thmath-phmath.MP
keywords multivariate hypergeometric functionsMathematica packagePfaffian systemnumerical evaluationanalytic continuationFeynman integralsAppell functionsLauricella functions
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The pith

HyperPrecision package evaluates general multivariate hypergeometric functions at high precision by reducing their Pfaffian PDE systems to solvable ODEs along contours.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a Mathematica package that computes numerical values of Horn-type multivariate hypergeometric functions and their expansions in a small parameter outside the limited domains where their defining series converge. It achieves this by automatically generating the associated Pfaffian system of partial differential equations, restricting that system to a one-dimensional contour connecting an initial point to the desired evaluation point, and integrating the resulting ordinary differential equation with the Frobenius method while setting boundary conditions from the series definition. A sympathetic reader would care because these functions appear in quantum field theory, string theory, and cosmology, where manual analytic continuation is often intractable and high-precision results are required. The package demonstrates the procedure on Appell, Horn, and Lauricella functions together with applications to Feynman integrals and correlators.

Core claim

The central claim is that any general Horn-type multivariate hypergeometric function admits an automatically constructible Pfaffian system whose restriction to a suitable one-dimensional contour yields an ordinary differential equation that the Frobenius method can integrate numerically to arbitrary precision once the boundary conditions are fixed analytically by the defining series.

What carries the argument

The Pfaffian system of partial differential equations for the hypergeometric function, restricted to a one-dimensional contour in variable space and integrated by the Frobenius method with series-supplied initial conditions.

If this is right

  • High-precision numerical values become available for the Appell F1, F2, F3, and F4 functions at arbitrary points in their domains.
  • The same procedure supplies values for the Horn G- and H-series and the Lauricella FA, FB, FC, and FD functions.
  • Laurent expansions in a small parameter ε can be obtained simultaneously with the function values themselves.
  • Angular integrals, Feynman integrals, and cosmological or holographic correlators that reduce to these hypergeometric functions can be evaluated numerically to high precision.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The automatic Pfaffian construction could be reused as a building block for numerical continuation of other systems of linear PDEs that arise in physics but lack closed-form continuations.
  • Integration of the package into existing Feynman-integral toolchains would allow direct high-precision checks against sector-decomposition or other numerical methods without intermediate analytic steps.
  • The contour-reduction technique suggests that similar one-dimensional paths might be identified for multivariate functions outside the Horn class when their differential equations are known.

Load-bearing premise

A suitable one-dimensional contour can always be chosen so that the reduced ordinary differential equation encounters neither prohibitive singularities nor insurmountable numerical instabilities.

What would settle it

A concrete counterexample would be any specific Horn-type function together with a target point for which the package either fails to construct the Pfaffian system or produces an ODE whose numerical integration diverges or becomes unstable on every possible contour.

Figures

Figures reproduced from arXiv: 2605.30216 by Souvik Bera, Sumit Banik.

Figure 1
Figure 1. Figure 1: Random sampling of (x, y) points used to validate the ϵ-expansion in Eq. (5.1) of Appell F2 against PrecisionLauricella. The points we checked are shown as green dots, with six points highlighted in red corresponding to the points given in [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: N-loop banana integral In this subsection, we consider the N-loop banana Feynman integral ( [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Random sampling of kinematic points (m2 1 , p2 ) used to validate the one-loop bubble integral in Eq. (5.15) against the Appell F4 representation in Eq. (5.16). The kinematic points we checked are shown as green dots, with six points highlighted in red corresponding to the points given in [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evaluation time of HyperPrecision for the Lauricella functions F (n) A , F (n) B , and F (n) D as a function of the number of variables n. The observed time scaling in [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The normalised bispectrum shape function [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The normalised bispectrum shape function [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The analytically continued three-point function [PITH_FULL_IMAGE:figures/full_fig_p032_7.png] view at source ↗
read the original abstract

In this paper, we present HyperPrecision, a Mathematica package for high-precision numerical evaluation of general Horn-type multivariate hypergeometric functions and their Laurent expansions in a small parameter $\epsilon$. Such functions appear widely in physics and mathematics, with applications ranging from quantum field theory and string theory to number theory and statistics. Their high-precision numerical evaluation, however, remains challenging, since their defining series converge only in restricted domains and analytic continuation beyond these domains is, in general, non-trivial. HyperPrecision addresses this problem by automatically constructing the Pfaffian system of partial differential equations for a given hypergeometric function and restricting it to a one-dimensional contour in the space of variables connecting the starting to the target point. The resulting ordinary differential equation is then solved by the Frobenius method, with boundary conditions fixed analytically by the defining series. We illustrate the use of the package by evaluating commonly occurring multivariate hypergeometric functions, including the Appell $F_1$, $F_2$, $F_3$, and $F_4$ functions, the Horn $G$- and $H$-series, and the Lauricella $F_A$, $F_B$, $F_C$, and $F_D$ functions, as well as by considering applications to angular integrals, Feynman integrals, and cosmological and holographic correlators.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper presents HyperPrecision, a Mathematica package for high-precision numerical evaluation of general Horn-type multivariate hypergeometric functions and their Laurent expansions in a small parameter ε. The approach automatically constructs the Pfaffian system of partial differential equations for a given function, restricts it to a one-dimensional contour connecting starting and target points, solves the resulting ODE via the Frobenius method, and fixes boundary conditions analytically from the defining series. Illustrations cover Appell F1–F4, Horn G- and H-series, Lauricella FA–FD functions, plus applications to angular integrals, Feynman integrals, and cosmological/holographic correlators.

Significance. If the automated Pfaffian construction and contour selection prove reliable beyond the listed families, the package would provide a valuable general-purpose tool for high-precision computations in quantum field theory, string theory, and related fields where multivariate hypergeometrics arise. Shipping a Mathematica package with code for the described pipeline is a concrete strength that supports reproducibility.

major comments (2)
  1. [Abstract and algorithmic description] Abstract (and algorithmic sections): the claim that a Pfaffian system can be constructed automatically for arbitrary Horn-type functions, and that a suitable one-dimensional contour can always be chosen so the restricted ODE has only regular, avoidable singularities, lacks a general existence proof or systematic failure-mode analysis for functions outside the standard Appell/Horn/Lauricella families; this is load-bearing because the entire numerical pipeline collapses if either step fails for a non-standard series.
  2. [Validation and benchmarks] Validation sections: no detailed benchmarks, error analysis, or singularity-handling tests are supplied that would confirm the claimed high-precision results hold across the full scope of Horn-type functions; without these the soundness of the high-precision claim cannot be verified.
minor comments (1)
  1. Notation for the Frobenius-method implementation and contour parametrization could be made more explicit to aid readers implementing or extending the package.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to strengthen the presentation of the algorithmic scope and to include additional validation material.

read point-by-point responses

  1. Referee: Abstract (and algorithmic sections): the claim that a Pfaffian system can be constructed automatically for arbitrary Horn-type functions, and that a suitable one-dimensional contour can always be chosen so the restricted ODE has only regular, avoidable singularities, lacks a general existence proof or systematic failure-mode analysis for functions outside the standard Appell/Horn/Lauricella families; this is load-bearing because the entire numerical pipeline collapses if either step fails for a non-standard series.

    Authors: We agree that the manuscript does not contain a general existence theorem covering every conceivable Horn-type series. The Pfaffian construction implemented in HyperPrecision follows the standard algorithmic procedure based on the coefficient recurrence relations of a given Horn series (as described in the cited literature on multivariate hypergeometric systems). The package therefore applies to any series for which such a Pfaffian system exists and can be computed symbolically; it does not claim to handle pathological cases outside this class. In the revised version we have added an explicit subsection (Section 3.2) that states the working assumptions, lists the families for which the construction has been verified, and describes the failure modes we have encountered (e.g., when the recurrence relations do not close into a finite Pfaffian system or when all admissible contours encounter unavoidable irregular singularities). We have also inserted a short paragraph in the abstract clarifying that the method is constructive for the class of Horn-type functions admitting a Pfaffian representation. revision: yes

  2. Referee: Validation sections: no detailed benchmarks, error analysis, or singularity-handling tests are supplied that would confirm the claimed high-precision results hold across the full scope of Horn-type functions; without these the soundness of the high-precision claim cannot be verified.

    Authors: We accept that the original validation material was limited to illustrative examples. In the revised manuscript we have added a dedicated numerical-validation section (Section 5) containing: (i) systematic high-precision comparisons against known reference values for Appell F1–F4, Horn G/H, and Lauricella FA–FD functions over a grid of kinematic points, (ii) an error analysis that reports absolute and relative errors as functions of working precision and contour length, and (iii) explicit singularity-handling tests that approach regular singular points along several contours while monitoring the achieved precision. All benchmarks were performed with Mathematica’s arbitrary-precision arithmetic and are accompanied by the corresponding notebook files that will be deposited with the package. revision: yes

Circularity Check

0 steps flagged

No circularity: method derives ODEs directly from series definitions

full rationale

The paper describes an algorithmic pipeline that starts from the defining multivariate series of Horn-type hypergeometric functions, constructs the associated Pfaffian system of PDEs, restricts it to a contour, and solves the resulting ODE via the Frobenius method with series-derived boundary conditions. This chain is self-contained and does not reduce any claimed output to a fitted parameter, self-citation, or renamed input; the construction steps are presented as operating on the function definitions themselves. No load-bearing uniqueness theorem, ansatz, or prediction is invoked that collapses to the inputs by construction. The approach therefore qualifies as an independent implementation rather than a circular re-derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of hypergeometric functions and their associated differential systems; no new physical axioms, free parameters, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Pfaffian systems exist and can be constructed for general Horn-type multivariate hypergeometric functions.
    Invoked when the package automatically builds the system of PDEs from the function definition.
  • standard math The Frobenius method can be applied to the reduced ODE along a chosen contour with boundary conditions from the defining series.
    Used to solve the resulting ordinary differential equation numerically.

pith-pipeline@v0.9.1-grok · 5774 in / 1578 out tokens · 33553 ms · 2026-06-29T06:28:48.096162+00:00 · methodology

discussion (0)

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Reference graph

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