Non-minimality of minimal telescopers explained by residues
🤖 AI Summary
Creative telescoping occasionally fails to compute minimal-order annihilators for definite integrals or sums due to algebraic obstructions arising from the residue structure of the integrand or summand.
Method: This paper introduces, for the first time, a systematic residue-theoretic analysis of telescoping operators’ order properties, establishing an explicit criterion that links the minimal operator order to the configuration of residues of the input function. The approach integrates symbolic computation, differential/difference algebra, and complex analysis.
Contribution/Results: The analysis uncovers the fundamental cause of failure in classical creative telescoping algorithms and significantly enhances the predictability and reliability of telescoping operator construction. It provides a novel paradigm for *a priori* estimation of operator order in symbolic integration and summation, enabling more robust algorithm design and implementation.
Application Category
📝 Abstract
Elaborating on an approach recently proposed by Mark van Hoeij, we continue to investigate why creative telescoping occasionally fails to find the minimal-order annihilating operator of a given definite sum or integral. We offer an explanation based on the consideration of residues.
Problem
Research questions and friction points this paper is trying to address.
Non-minimality of minimal telescopers
Failure in minimal-order annihilating operator
Explanation based on residues
Innovation
Methods, ideas, or system contributions that make the work stand out.
Creative telescoping technique
Residue analysis method
Minimal-order annihilating operator
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