Telescopers for differential forms with one parameter

TL;DR

This paper extends the concept of telescopers to differential forms with D-finite coefficients, providing conditions for their existence and algorithms for their computation, with applications in various mathematical fields.

Contribution

It introduces telescopers for differential forms with D-finite coefficients, establishing existence criteria and computational methods, expanding their use in mathematics.

Findings

Established necessary and sufficient conditions for telescopers of differential forms.

Developed algorithms to verify the existence of such telescopers.

Provided methods to compute telescopers when they exist.

Abstract

Telescopers for a function are linear differential (resp. difference) operators annihilated by the definite integral (resp. definite sum) of this function. They play a key role in Wilf-Zeilberger theory and algorithms for computing them have been extensively studied in the past thirty years. In this paper, we introduce the notion of telescopers for differential forms with $D$-finite function coefficients. These telescopers appear in several areas of mathematics, for instance parametrized differential Galois theory and mirror symmetry. We give a sufficient and necessary condition for the existence of telescopers for a differential form and describe a method to compute them if they exist. Algorithms for verifying this condition are also given.