Creative Telescoping on Multiple Sums

TL;DR

This paper presents practical strategies using creative telescoping to analyze complex triple binomial sums arising from quasi-Monte Carlo methods, highlighting challenges and the importance of certificates in computations.

Contribution

It introduces new approaches and discusses difficulties in automating creative telescoping for multiple sums, emphasizing the role of certificates and computational complexity.

Findings

Strategies for handling triple binomial sums

Challenges in automating creative telescoping

Impact of certificates on computation time

Abstract

We showcase a collection of practical strategies to deal with a problem arising from an analysis of integral estimators derived via quasi-Monte Carlo methods. The problem reduces to a triple binomial sum, thereby enabling us to open up the holonomic toolkit, which contains tools such as creative telescoping that can be used to deduce a recurrence satisfied by the sum. While applying these techniques, a host of issues arose that partly needed to be resolved by hand. In other words, no creative telescoping implementation currently exists that can resolve all these issues automatically. Thus, we felt the need to compile the different strategies we tried and the difficulties that we encountered along the way. In particular, we highlight the necessity of the certificate in these computations and how its complexity can greatly influence the computation time.