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HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters

T. Huber, D. Maitre

TL;DR

Hypergeometric functions with integer-valued parameters arise in dimensional regularization of loop and phase-space integrals. The paper introduces HypExp, a Mathematica package that automates systematic ε-expansions around integers using two complementary strategies: an integration-based method focused on $_2F_1$ via integral representations and a nested-sums approach grounded in Z/S sums and their connections to harmonic polylogarithms. It also develops new integral classes, handles unit-argument cases, and provides gamma-function expansions and polylogarithm relations, with a library mechanism to cache and reuse results. This tool enables high-order, symbolic expansions essential for precise multi-loop calculations in quantum field theory.

Abstract

We present the Mathematica package HypExp which allows to expand hypergeometric functions $_JF_{J-1}$ around integer parameters to arbitrary order. At this, we apply two methods, the first one being based on an integral representation, the second one on the nested sums approach. The expansion works for both symbolic argument $z$ and unit argument. We also implemented new classes of integrals that appear in the first method and that are, in part, yet unknown to Mathematica.

HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters

TL;DR

Hypergeometric functions with integer-valued parameters arise in dimensional regularization of loop and phase-space integrals. The paper introduces HypExp, a Mathematica package that automates systematic ε-expansions around integers using two complementary strategies: an integration-based method focused on via integral representations and a nested-sums approach grounded in Z/S sums and their connections to harmonic polylogarithms. It also develops new integral classes, handles unit-argument cases, and provides gamma-function expansions and polylogarithm relations, with a library mechanism to cache and reuse results. This tool enables high-order, symbolic expansions essential for precise multi-loop calculations in quantum field theory.

Abstract

We present the Mathematica package HypExp which allows to expand hypergeometric functions around integer parameters to arbitrary order. At this, we apply two methods, the first one being based on an integral representation, the second one on the nested sums approach. The expansion works for both symbolic argument and unit argument. We also implemented new classes of integrals that appear in the first method and that are, in part, yet unknown to Mathematica.

Paper Structure

This paper contains 25 sections, 97 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Method of integration
  3. $_2F_1$-algorithm
  4. Expansion of the basic $_2F_1$-functions
  5. Integrals and their algorithm
  6. General aspects
  7. Description of the algorithm
  8. Integrals of unit argument
  9. Nested sums method
  10. Definitions and auxilary functions
  11. Relations to other functions
  12. Expansion of the $\Gamma$ function
  13. Description of the algorithm
  14. Last summation
  15. Hypergeometric functions of unit argument
  16. ...and 10 more sections

Figures (2)

  • Figure 1: $_2F_1$-algorithm for $b_1 = -1$
  • Figure 2: $_2F_1$-algorithm for $b_1 \ge 0$