Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation

TL;DR

This paper reviews difference ring theory for symbolic summation, focusing on canonical representations and their implementation in the Sigma summation package, enhancing the clarity and usability of symbolic summation tools.

Contribution

It introduces a detailed framework for canonical representations in difference rings and their integration into the Sigma package, improving symbolic summation methods.

Findings

Canonical representations are uniquely characterized within the term algebra.

The paper provides precise input-output specifications for Sigma tools.

Enhanced translation and back translation processes are established.

Abstract

A general overview of the existing difference ring theory for symbolic summation is given. Special emphasis is put on the user interface: the translation and back translation of the corresponding representations within the term algebra and the formal difference ring setting. In particular, canonical (unique) representations and their refinements in the introduced term algebra are explored by utilizing the available difference ring theory. Based on that, precise input-output specifications of the available tools of the summation package Sigma are provided.