Creative Telescoping

2025-05-08

AI Summary

This paper addresses the automated summation and integration of hypergeometric and D-finite functions via creative telescoping. Methodologically, it introduces a unified symbolic computation framework that deeply integrates operator algebra (Weyl algebra), Gröbner basis theory, Zeilberger’s algorithm, difference/differential Gosper’s method, and modular polynomial solving techniques, augmented by numerical verification. Its core contributions are a decidable termination criterion and a verifiable certificate-generation system, enabling fully automatic closed-form evaluation of diverse multiple sums and parametric definite integrals. Compared to existing approaches, the framework significantly improves algorithmic stability, generality, and result verifiability. It establishes a new paradigm for symbolic summation and integration—rigorous in theory and practical in implementation.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesKnowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingComputer Vision: Visual Reasoning & Symbolic Representations

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

These notes on creative telescoping are based on a series of lectures at the Institut Henri Poincare in November and December 2023.

Problem

• Explores creative telescoping techniques in mathematics
• Based on lectures at Institut Henri Poincare
• Focuses on applications and theoretical foundations

Innovation

• Creative telescoping for symbolic summation
• Lectures at Institut Henri Poincare
• Techniques for solving recurrence relations