Telescoping Algorithms for $\Sigma^*$-Extensions via Complete Reductions

TL;DR

This paper introduces a new complete reduction framework for $ abla$-extensions in difference fields, enhancing the efficiency of parameterized telescoping and sum depth reduction in symbolic summation.

Contribution

It develops a complete reduction method for $ abla$-extensions, improving telescoping algorithms and connecting to Karr's theorem for better sum analysis.

Findings

Efficient framework for parameterized telescoping in $ abla$-extensions.

Structural theorems linking complete reductions to Karr's theorem.

Reduction of sum depth in combinatorial and physical applications.

Abstract

A complete reduction on a difference field is a linear operator that enables one to decompose an element of the field as the sum of a summable part and a remainder such that the given element is summable if and only if the remainder is equal to zero. In this paper, we present a complete reduction in a tower of $\Sigma^*$-extensions that turns to a new efficient framework for the parameterized telescoping problem. Special instances of such $\Sigma^*$-extensions cover iterative sums such as the harmonic numbers and generalized versions that arise, e.g., in combinatorics, computer science or particle physics. Moreover, we illustrate how these new ideas can be used to reduce the depth of the given sum and provide structural theorems that connect complete reductions to Karr's Fundamental Theorem of symbolic summation.