Formal residue and computer proofs of combinatorial identities

TL;DR

This paper introduces a method using formal residues and an extended Zeilberger's algorithm to generate recurrence relations, enabling computer proofs of combinatorial identities and the discovery of new ones.

Contribution

It presents a novel approach combining formal residues with an extended Zeilberger's algorithm for automated proof and discovery of combinatorial identities.

Findings

Successfully proved several known combinatorial identities using computer algebra.

Derived new identities involving Stirling numbers.

Studied the applicability and limitations of the proposed method.

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'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.