How to generate all possible rational Wilf-Zeilberger forms?

TL;DR

This paper provides a comprehensive structural description of all rational Wilf-Zeilberger forms, extending the classical Ore-Sato theorem to a multivariate additive setting, aiding in combinatorial identity proofs.

Contribution

It introduces a structural characterization of rational Wilf-Zeilberger forms and a decomposition of multivariate hyperarithmetic terms, expanding the theoretical framework.

Findings

Structural description of all rational Wilf-Zeilberger forms

Additive analog of the Ore-Sato theorem

Decomposition of multivariate hyperarithmetic terms

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.