  The coefficient of x^{-1} of a formal Laurent series f(x) is called the
formal residue of f(x). Many combinatorial numbers can be represented by the
formal residues of hypergeometric terms. With these representations and the
extended Zeilberger's algorithm, we generate recurrence relations for
summations involving combinatorial sequences such as Stirling numbers. As
examples, we give computer proofs of several known identities and derive some
new identities. The applicability of this method is also studied.
