Symbolic Computation Group

Wilf and Zeilberger conjectured in 1992 that a hypergeometric term is proper-hypergeometric if and only if it is holonomic. Their conjecture concerns hypergeometric terms which depend on several discrete and continuous variables. Jointly with S.A. Abramov, we have proved a slightly modified version of their conjecture in the multivariate discrete case, namely that every holonomic hypergeometric term is conjugate to a proper term (meaning that they have the same certificates). Our proof is based on the Ore-Sato Theorem which states essentially that for every hypergeometric term T there is a rational function R and a proper term T' such that T and RT' have the same certificates. This was proved in the bivariate case by Ore using elementary means, and in the multivariate case by Sato using homological algebra. We give an elementary proof of the multivariate Ore-Sato Theorem. The necessary tools that are useful also for other purposes are normal forms of rational functions and several notions of shift-invariance for multivariate polynomials. We have also shown that a rational sequence is holonomic if and only if its denominator factors into integer-linear factors over the algebraic closure of the coefficient field.