Creative Telescoping
Shaoshi Chen, Manuel Kauers, Christoph Koutschan
TL;DR
This survey presents creative telescoping as a unifying framework for deriving recurrences and differential equations for parametric sums and integrals, spanning hypergeometric, rational, and D-finite settings. It integrates classical methods (Gosper, Sister Celine, Zeilberger) within the holonomic/D-finite paradigm and extends to multivariate problems via Abramov–van Hoeij and Chyzak algorithms, with a focus on practical computation and closure properties. The text highlights mechanism-like tools such as telescopers $P$, certificates $Q$, residues, diagonals, and Hadamard products, and showcases software implementations and real-world applications in combinatorics, physics, and geometry. Overall, it illuminates how differential- and difference-field techniques complement each other and points to future directions for a unified theory and more efficient algorithms.
Abstract
These notes on creative telescoping are based on a series of lectures at the Institut Henri Poincare in November and December 2023.