Formal residue and computer proofs of combinatorial identities

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.