Proof of the Wilf-Zeilberger Conjecture for Mixed Hypergeometric Terms

TL;DR

This paper extends the Wilf-Zeilberger conjecture to mixed hypergeometric terms involving both discrete and continuous variables, proving a conjugate interpretation that generalizes previous discrete-only results.

Contribution

It provides the first proof of the conjugate interpretation of the Wilf-Zeilberger conjecture for mixed hypergeometric terms with discrete and continuous variables.

Findings

Proves the conjugate interpretation in the mixed setting.

Extends previous discrete-only results to mixed variables.

Establishes a theoretical foundation for holonomic and proper hypergeometric terms in mixed cases.

Abstract

In 1992, Wilf and Zeilberger conjectured that a hypergeometric term in several discrete and continuous variables is holonomic if and only if it is proper. Strictly speaking the conjecture does not hold, but it is true when reformulated properly: Payne proved a piecewise interpretation in 1997, and independently, Abramov and Petkovsek in 2002 proved a conjugate interpretation. Both results address the pure discrete case of the conjecture. In this paper we extend their work to hypergeometric terms in several discrete and continuous variables and prove the conjugate interpretation of the Wilf-Zeilberger conjecture in this mixed setting.