New Bounds for Hypergeometric Creative Telescoping

TL;DR

This paper introduces improved bounds for the order of telescopers in hypergeometric creative telescoping, providing a new proof of termination and enhancing understanding of the algorithm's efficiency.

Contribution

It offers a new termination proof and derives tighter bounds for telescoper orders in hypergeometric terms, improving upon existing literature.

Findings

New bounds sometimes better than existing ones

Provides an independent proof of algorithm termination

Derives both lower and upper bounds for telescoper order

Abstract

Based on a modified version of Abramov-Petkov\v{s}ek reduction, a new algorithm to compute minimal telescopers for bivariate hypergeometric terms was developed last year. We investigate further in this paper and present a new argument for the termination of this algorithm, which provides an independent proof of the existence of telescopers and even enables us to derive lower as well as upper bounds for the order of telescopers for hypergeometric terms. Compared to the known bounds in the literature, our bounds are sometimes better, and never worse than the known ones.