On rational definite summation

TL;DR

This paper offers a partial proof of a conjecture related to the algorithmic computation of finite sums of rational functions, providing an algorithm for a broad class of such sums.

Contribution

It presents a partial proof of the van Hoeij-Abramov conjecture and introduces an algorithm for computing sums of rational functions.

Findings

Provides a partial proof of the van Hoeij-Abramov conjecture.

Develops an algorithm for summing a large class of rational functions.

Advances the theoretical understanding of rational summation algorithms.

Abstract

We present a partial proof of van Hoeij-Abramov conjecture about the algorithmic possibility of computation of finite sums of rational functions. The theoretical results proved in this paper provide an algorithm for computation of a large class of sums $ S(n) = \sum_{k=0}^{n-1}R(k,n)$.