Wilf–Zeilberger conjecture

Wilf–Zeilberger conjecture
Name	Wilf–Zeilberger conjecture
Field	Mathematics
Subfield	Combinatorics; Symbolic computation; Special functions
Formed	1990s
Notable	Doron Zeilberger; Herbert Wilf; Marko Petkovšek; Manuel Kauers; Christoph Koutschan

Wilf–Zeilberger conjecture is a conjecture in Mathematics proposing a structural characterization of hypergeometric and holonomic multivariate terms that admit certificate-based proofs and algorithmic summation. It arose from collaborations and exchanges involving Herbert Wilf, Doron Zeilberger, Marko Petkovšek, and researchers in computer algebra and combinatorics such as Manuel Kauers and Christoph Koutschan. The conjecture stimulated developments linking algorithmic tools like creative telescoping and the Gosper algorithm to structural results in the theory of special functions and difference equations.

History and statement of the conjecture

The conjecture was formulated in the context of the 1990s surge in algorithmic proof theory driven by figures like Herbert Wilf, Doron Zeilberger, Gosper, R. W., Marko Petkovšek, and communities around conferences such as ISSAC and workshops at institutions including Massachusetts Institute of Technology and University of Pennsylvania. Early expositions connected to the conjecture referenced algorithmic breakthroughs such as Gosper's algorithm, the Zeilberger algorithm, and results by Petkovšek on hypergeometric term solutions. The statement asserts that a multivariate hypergeometric term admits a so-called Wilf–Zeilberger pair (a certificate giving a finite summation identity) precisely when the term is of a specific structural form related to products of rational functions and factorial-like factors; this connects to work by Stanley, Gessel, and Andrews on q-series and orthogonal polynomials. Subsequent presentations by researchers at RISC and MPI MiS clarified formal hypotheses and motivated refined versions.

Definitions and preliminaries

Key definitions use terminology introduced by Gosper, Zeilberger, Petkovšek, and Stanley: a hypergeometric term is a function whose termwise ratios are rational functions, a holonomic function satisfies linear recurrences with polynomial coefficients, and a certificate is an auxiliary function producing a telescoping relation. Preliminary machinery involves difference equations studied by E. T. Whittaker and George Boole historically, algorithmic frameworks like creative telescoping developed by Zeilberger and formalizations in Ore algebra originating with Oystein Ore. The definitions also invoke notions from the theory of D-finite functions advanced by André Chyzak and the structural tools used by Koutschan and Chyzak for multivariate summation.

Known cases and proofs

Several special cases of the conjecture were established by targeted methods: univariate hypergeometric summation results trace to Gosper and Zeilberger; bivariate rational cases were resolved using techniques by Abramov and Petkovšek; hyperexponential analogues were addressed by Harris and work in symbolic integration communities including Risch algorithm researchers. Algorithmic proofs exploiting Ore algebras and annihilators were given by Chyzak, Koutschan, and Kauers for classes of holonomic functions; independent confirmations used constructive approaches from Wilf and Zeilberger's school. Computational evidence involving implementations in Maple, Mathematica, and packages developed at RISC and Inria further established many instances.

Counterexamples and refinements

Counterexamples to naive formulations were produced by constructions related to pathological multivariate rational terms and by leveraging results from algebraic geometry contributors such as Jean-Pierre Serre and Alexander Grothendieck in the guise of obstructions to factorization. These led researchers including Kauers, Koutschan, and Chen to propose refined statements incorporating conditions on the singularity structure, denominator factorization, and the presence of mixed shift behaviors. Work by Bostan, Chyzak, and Singer clarified necessary hypotheses, while negative results motivated alternate conjectures and the addition of constraints inspired by the theory of difference Galois theory developed by Michael Singer and others.

Applications and consequences

The conjecture and its developments influenced automated proof systems used in combinatorial identities encountered in work by Andrews, Gasper, Wilf, and Zeilberger, with concrete applications to summation identities in q-series, hypergeometric functions, and identities arising in enumerative problems studied by Richard Stanley, Ira Gessel, and Philippe Flajolet. Consequences include optimized algorithms in symbolic summation implemented in systems at Wolfram Research and Maplesoft, improvements in verification of identities used in mathematical physics contexts involving contributors like Barry McCoy and F. Y. Wu, and impacts on algorithmic complexity analyses by researchers at INRIA and TU Graz.

Connections to creative telescoping and symbolic summation

The conjecture sits at the heart of creative telescoping theory as formalized by Zeilberger, with algorithmic realizations by Chyzak, Koutschan, and Bostan linking to implementations in Maple and Mathematica. It motivated developments in symbolic summation algorithms like the Zeilberger algorithm, extensions by Chyzak to mixed cases, and the systematic use of Ore algebra frameworks advanced by Bronstein and Singer. Links to broader algorithmic paradigms appear in work by Gessel, Stanley, Petkovšek, and Wilf who influenced curricula and software at institutions such as MIT, Princeton University, and Johannes Kepler University Linz.

Category:Conjectures in mathematics