A Difference Ring Theory for Symbolic Summation

TL;DR

This paper introduces an advanced difference ring framework that extends Karr's approach, enabling automated symbolic summation of complex nested sums and products, with applications in combinatorics and physics.

Contribution

It develops a comprehensive difference ring theory with algorithms for constructing difference rings and solving summation problems, including telescoping and creative telescoping.

Findings

Algorithms successfully solve parameterized difference equations.

Framework handles nested sums and products over roots of unity.

Applied to complex problems in combinatorics and particle physics.

Abstract

A summation framework is developed that enhances Karr's difference field approach. It covers not only indefinite nested sums and products in terms of transcendental extensions, but it can treat, e.g., nested products defined over roots of unity. The theory of the so-called $R\Pi\Sigma^*$-extensions is supplemented by algorithms that support the construction of such difference rings automatically and that assist in the task to tackle symbolic summation problems. Algorithms are presented that solve parameterized telescoping equations, and more generally parameterized first-order difference equations, in the given difference ring. As a consequence, one obtains algorithms for the summation paradigms of telescoping and Zeilberger's creative telescoping. With this difference ring theory one obtains a rigorous summation machinery that has been applied to numerous challenging problems coming, e.g., from combinatorics and particle physics.