Computerized proofs of hypergeometric identities: Methods, advances, and limitations

In this thesis, we consider the impact of computers on the proof of identities in mathematics. We are primarily concerned with hypergeometric identities, which take on a form which is supremely suited for exploration with computers. We consider Sister Celine’s distinctly pre-computer algorithm, which served as the inspiration for the later algorithms we consider by Gosper and Zeilberger. Each of these three algorithms is designed to find a closed form solution of a hypergeometric summation. Following our exposition of these three algorithms, we consider the WZ method, a powerful application of Zeilberger’s algorithm which can be used to conclusively prove many known (or conjectured) hypergeometric identities. We also briefly explore added bonuses that come from the application of the WZ method. Next, we look at improvements and refinements both in the implementation of the algorithms themselves and the computer technology on which they are run. We also briefly discuss the advantages and disadvantages of the transition to computer proof and the impact of computer proof on human mathematicians --Abstract, page iii