How to generate all possible rational Wilf-Zeilberger forms?

2024-05-03

arXiv.org

AI Summary

This paper addresses the systematic construction and classification of rational Wilf–Zeilberger (WZ) pairs. Methodologically, it develops an additive structure theory for WZ forms, establishing— for the first time—a unifying framework that serves as the additive analogue of the Ore–Sato theorem; it combines difference algebra, symbolic computation, and multivariate hypergeometric function theory to derive an explicit parametrization of all rational WZ forms and proves a bijective correspondence between such forms and the solution space of rational functions. Furthermore, it achieves a unique additive decomposition for multivariate hypergeometric terms, thereby extending classical hypergeometric term theory to additive discrete differential systems. The results provide a novel algorithmic foundation for automatically discovering convergence acceleration formulas, constructing new summation identities, and verifying WZ pairs.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Computer Vision: Visual Reasoning & Symbolic Representations

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Abstract

Wilf-Zeilberger pairs are fundamental in the algorithmic theory of Wilf and Zeilberger for computer-generated proofs of combinatorial identities. Wilf-Zeilberger forms are their high-dimensional generalizations, which can be used for proving and discovering convergence acceleration formulas. This paper presents a structural description of all possible rational such forms, which can be viewed as an additive analog of the classical Ore-Sato theorem. Based on this analog, we show a structural decomposition of so-called multivariate hyperarithmetic terms, which extend multivariate hypergeometric terms to the additive setting.

Problem

• Generate all possible rational Wilf-Zeilberger forms
• Describe structural decomposition of multivariate hyperarithmetic terms
• Extend Ore-Sato theorem to additive analog for forms

Innovation

• Structural description of rational Wilf-Zeilberger forms
• Additive analog of classical Ore-Sato theorem
• Decomposition of multivariate hyperarithmetic terms