Multivariate Hypergeometric Terms

Abstract

In this 1997 Ph.D. dissertation we prove a piecewise form of the discrete part of Wilf and Zeilberger's 1992 conjecture that a hypergeometric term is proper if and only if it is holonomic. We show that a holonomic hypergeometric term on $Z^n$ is piecewise proper and we show that without such a qualification the conjecture is false. We call a term piecewise proper if $Z^n$ can be expressed as the union of a finite number of polyhedral regions (the "pieces") and a set of measure zero (which we define to be a finite union of hyperplanes) such that the restriction of the term to each polyhedral region is proper. We prove a similar result for terms that are not holonomic but honest. We call a term $h$ honest if for every vector $v$ in $Z^n$ there exist relatively prime polynomials $A_v$ and $B_v$ such that $A_v(z) h(z) = B_v(z) h(z+v)$ except on a set of measure zero. We also give a naive proof of the Ore--Sato Theorem using Gosper's Lemma. We solve an unrelated problem of Cameron by showing that there is a sum-free complete subset of $Z/mZ$ that is not symmetric for every sufficiently large modulus $m$, and we show that such a set must have the property that the cardinality of its sum set is greater than the cardinality of its difference set, which makes it a counterexample to a modular version of a conjecture of Conway. A set $S$ is said to be sum-free, complete, and symmetric respectively if $|S+S| \subset S^c$, $|S+S| \supset S^c$, and $S = -S$.