A streamlined difference ring theory: Indefinite nested sums, the alternating sign and the parameterized telescoping problem

TL;DR

This paper develops an algebraic framework using difference rings to represent nested sums, including alternating signs, and provides algorithms for solving telescoping problems, advancing symbolic summation techniques.

Contribution

It extends Karr's difference field theory to difference rings that incorporate the alternating sign, enabling more comprehensive symbolic summation.

Findings

Extended difference ring theory to include alternating signs.

Developed algorithms for parameterized telescoping solutions.

Enabled solving telescoping and creative telescoping problems in this framework.

Abstract

We present an algebraic framework to represent indefinite nested sums over hypergeometric expressions in difference rings. In order to accomplish this task, parts of Karr's difference field theory have been extended to a ring theory in which also the alternating sign can be expressed. The underlying machinery relies on algorithms that compute all solutions of a given parameterized telescoping equation. As a consequence, we can solve the telescoping and creative telescoping problem in such difference rings.