Reduction-Based Creative Telescoping for Definite Summation of D-finite Functions

TL;DR

This paper introduces a new algorithm for creative telescoping that efficiently computes telescopers and certificates for definite sums of D-finite functions, utilizing a discrete Hermite reduction and demonstrating practical performance.

Contribution

It presents a novel telescoping algorithm based on a discrete Hermite reduction, improving the computation of certificates for D-finite functions.

Findings

Efficient Maple implementation with good timings

Successful computation of telescopers and certificates for various examples

Advances the algorithmic approach to summation of D-finite functions

Abstract

Creative telescoping is an algorithmic method initiated by Zeilberger to compute definite sums by synthesizing summands that telescope, called certificates. We describe a creative telescoping algorithm that computes telescopers for definite sums of D-finite functions as well as the associated certificates in a compact form. The algorithm relies on a discrete analogue of the generalized Hermite reduction, or equivalently, a generalization of the Abramov-Petkov\v{s}ek reduction. We provide a Maple implementation with good timings on a variety of examples.