Order-Degree Curves for Hypergeometric Creative Telescoping

TL;DR

This paper establishes degree bounds for telescopers in hypergeometric creative telescoping, revealing a hyperbolic relationship between order and degree that guides the construction of recurrence operators.

Contribution

It introduces a novel hyperbolic curve-based bound for the degrees of telescopers, enhancing understanding of the order-degree trade-off in hypergeometric summation.

Findings

Bounds are expressed as hyperbolas in the (r,d)-plane.

Higher order telescopers tend to have lower degrees.

Provides a geometric perspective on degree bounds.

Abstract

Creative telescoping applied to a bivariate proper hypergeometric term produces linear recurrence operators with polynomial coefficients, called telescopers. We provide bounds for the degrees of the polynomials appearing in these operators. Our bounds are expressed as curves in the (r,d)-plane which assign to every order r a bound on the degree d of the telescopers. These curves are hyperbolas, which reflect the phenomenon that higher order telescopers tend to have lower degree, and vice versa.