Order Bounds for Hypergeometric and q-Hypergeometric Creative Telescoping

TL;DR

This paper develops a unified framework for computing minimal order telescopers for hypergeometric and q-hypergeometric terms, providing new bounds and proofs for their existence and order.

Contribution

It introduces a new argument for the termination of the telescoping algorithm, establishing unified bounds and confirming the existence of minimal order telescopers.

Findings

Unified bounds for hypergeometric and q-hypergeometric telescopers

Proof of algorithm termination and telescoper existence

Improved upper bounds in the q-hypergeometric case

Abstract

Leveraging a general framework adapted from symbolic integration, a unified reduction-based algorithm for computing telescopers of minimal order for hypergeometric and q-hypergeometric terms has been recently developed. In this paper, we conduct a deeper exploration and put forth a new argument for the termination of the algorithm. This not only provides an independent proof of existence of telescopers, but also allows us to derive unified upper and lower bounds on the order of telescopers for hypergeometric terms and their q-analogues. Compared with known bounds in the literature, our bounds, in the hypergeometric case, are exactly the same as the tight ones obtained in 2016; while in the q-hypergeometric case, no lower bounds were known before, and our upper bound is sometimes better and never worse than the known one.