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Simplifying Multiple Sums in Difference Fields

Carsten Schneider

TL;DR

Schneider surveys a difference-field framework for symbolic summation that unifies indefinite and definite multi-sum problems through $\\Pi\\Sigma^*$-fields and nested hypergeometric sums. The core workflow translates summands into a difference-field setting, solves telescoping and linear difference equations there, and then reinterprets the results as sums or sequences using an evaluation map; for definite sums, creative telescoping and recurrence solving are integrated and automated in the Sigma and EvaluateMultiSums packages. The work demonstrates automatic simplification to harmonic sums, multiple zeta values, and related generalizations, enabling large-scale multi-sum handling relevant to high-energy physics calculations. The practical impact is an extensible, automatic toolkit capable of transforming millions of multi-sums arising from Feynman integral computations into compact, analyzable expressions with rigorous correctness certificates.

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.

Simplifying Multiple Sums in Difference Fields

TL;DR

Schneider surveys a difference-field framework for symbolic summation that unifies indefinite and definite multi-sum problems through -fields and nested hypergeometric sums. The core workflow translates summands into a difference-field setting, solves telescoping and linear difference equations there, and then reinterprets the results as sums or sequences using an evaluation map; for definite sums, creative telescoping and recurrence solving are integrated and automated in the Sigma and EvaluateMultiSums packages. The work demonstrates automatic simplification to harmonic sums, multiple zeta values, and related generalizations, enabling large-scale multi-sum handling relevant to high-energy physics calculations. The practical impact is an extensible, automatic toolkit capable of transforming millions of multi-sums arising from Feynman integral computations into compact, analyzable expressions with rigorous correctness certificates.

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.

Paper Structure

This paper contains 13 sections, 5 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Indefinite summation: the basic mechanism
  3. Details of the difference field machinery
  4. Step 1: From indefinite nested sums and products to $\Pi\Sigma^*$-fields
  5. Step 2: Solving the telescoping problem in a given $\Pi\Sigma^*$-field
  6. Restriction to polynomial $\Pi\Sigma^*$-fields
  7. Step 3: Evaluating elements from a $\Pi\Sigma^*$-field to sequences
  8. Crucial property: algebraic independence of sequences
  9. The symbolic summation toolbox of Sigma
  10. Simplification of nested hypergeometric sum expressions
  11. Finding recurrence relations for definite sums
  12. Solving recurrence relations
  13. Simplification of multiple sums with EvaluateMultiSums

Key Result

lemma 1

Let $\mathbb{F}(t)$ be a rational function field, ${\sigma:\mathbb{F} \to \mathbb{F}}$ be a field automorphism, and $a,b\in\mathbb{F}$ with $a\neq0$. Then there is exactly one way how the field automorphism is extended to $\mathbb{F}(t)$ subject to the relation $\sigma(t)=a\,t+b$. Namely, for $f=\su

Theorems & Definitions (18)

  • definition 1
  • lemma 1
  • definition 2
  • definition 3
  • definition 4
  • remark 1
  • definition 5
  • theorem 1
  • definition 6
  • theorem 2
  • ...and 8 more