Symbolic Expansion of Transcendental Functions

TL;DR

This work presents nestedsums, a C++ library implemented with GiNaC to perform systematic Laurent expansions of higher transcendental functions that arise in quantum field theory, particularly in Feynman-integral calculations with a small parameter $\varepsilon$. It builds on the Hopf algebra of nested sums, introducing $Z$-sums and $S$-sums and integrating four generic expansion types (Type A–D) to reduce products to single sums, while handling Gamma-function ratios to manage constants. The library provides a robust set of object-oriented classes for letters, nested sums, and Gamma ratios, along with interface classes that map to generalized hypergeometric-type functions (Appell and Kampé de Fériet variants), and includes installation, documentation, interactive use, and a comprehensive checks-and-performance suite. Practical examples demonstrate hypergeometric and Appell-function expansions, with validation against known results and practical guidance for deploying the library in high-energy physics computations. The work thus offers a scalable, extensible tool for precise analytic expansions of transcendental functions in perturbative calculations, enabling more efficient and reliable extraction of Laurent coefficients in $\varepsilon$.

Abstract

Higher transcendental function occur frequently in the calculation of Feynman integrals in quantum field theory. Their expansion in a small parameter is a non-trivial task. We report on a computer program which allows the systematic expansion of certain classes of functions. The algorithms are based on the Hopf algebra of nested sums. The program is written in C++ and uses the GiNaC library.

Figures (2)

• Figure 1: The inheritance diagram for the class basic letter and its derived classes.
• Figure 2: The inheritance diagram for $Z$-sums and derived classes.