Formal residue and computer-assisted proofs of combinatorial identities

Authors : Jin HAITAO

Pages : 2466-2480

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :The coefficient of $x^{-1}$ of a formal Laurent series $finsert ignore into journalissuearticles values(x);$ is called the formal residue of $finsert ignore into journalissuearticles values(x);$. Many combinatorial numbers can be represented by the formal residues of hypergeometric terms. With these representations and the extended Zeilberger algorithm, we generate recurrence relations for summations involving combinatorial sequences such as Stirling numbers and their $q$-analog. As examples, we give computer proofs of several known identities and derive some new identities. The applicability of this method is also studied.
Keywords : Formal residue, extended Zeilberger algorithm, Stirling number