  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 $S = -S$.
