Simplifying Multiple Sums in Difference Fields

TL;DR

This paper reviews difference field algorithms for symbolic summation, emphasizing new methods for rephrasing and solving summation problems, and introduces software tools that automate complex calculations such as Feynman diagram evaluations.

Contribution

It introduces new difference field techniques for summation problems and combines them into software packages for automatic large-scale computations.

Findings

Discovered new harmonic number identities.

Extended summation techniques to Feynman diagram evaluations.

Automated summation for quantum chromodynamics calculations.

Abstract

In this survey article we present difference field algorithms for symbolic summation. Special emphasize is put on new aspects in how the summation problems are rephrased in terms of difference fields, how the problems are solved there, and how the derived results in the given difference field can be reinterpreted as solutions of the input problem. The algorithms are illustrated with the Mathematica package \SigmaP\ by discovering and proving new harmonic number identities extending those from (Paule and Schneider, 2003). In addition, the newly developed package \texttt{EvaluateMultiSums} is introduced that combines the presented tools. In this way, large scale summation problems for the evaluation of Feynman diagrams in QCD (Quantum ChromoDynamics) can be solved completely automatically.