A criterion for the applicability of Zeilberger's algorithm to rational functions

We consider the applicability (or terminating condition) of the well-known Zeilberger’s algorithm and give the complete solution to this problem for the case where the original hypergeometric term F(n; k) is a rational function. We specify a class of identities ∑n k=0 F(n; k) = 0; F(n; k)∈C(n; k), that cannot be proven by Zeilberger’s algorithm. Additionally, we give examples showing that the set of hypergeometric terms on which Zeilberger’s algorithm terminates is a proper subset of the set of all hypergeometric terms, but a super-set of the set of proper terms.