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. | Turk J Math (2018) 42: 2466 – 2480 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Formal residue and computer-assisted proofs of combinatorial identities Hai-Tao JIN∗, School of Science, Tianjin University of Technology and Education, Tianjin, . China Received: • Accepted/Published Online: • Final Version: Abstract: 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 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. Key words: Formal residue, extended Zeilberger algorithm, Stirling number 1. Introduction Finding recurrence relations for summations is a key step in computer proofs of combinatorial identities. In the 1990s, Wilf and Zeilberger [20, 21] developed the method of creative telescoping to generate recurrence relations for hypergeometric summations. Since then, many extensions and new algorithms have been discovered and designed for various kinds of summations. See, for example, [3, 5] for holonomic sequences, [1, 19] for multivariable hypergeometric terms, [17] for nested sums and products, [11, 12] for Stirling-like numbers, and [4, 13, 14] for nonholonomic sequences. Our approach is motivated by the work of Chen and Sun [2]. By using the Cauchy contour integral representations, they transformed sums involving Bernoulli numbers into hypergeometric summations. Then the recurrence relations for the sums can be derived by the extended Zeilberger algorithm [1]. In the present paper, we combine the formal residue operator .
