Deconstructing the Zeilberger algorithm

By looking under the hood of Zeilberger’s algorithm, as simplified by Mohammed and Zeilberger, it is shown that all the classical hypergeometric closed-form evaluations can be discovered ab initio, as well as many “strange” ones of Gosper, Maier, and Gessel and Stanton. The accompanying Maple package FindHypergeometric explains the various miracles that account for the classical evaluations, and the more specialized Maple package twoFone, also accompanying this article, finds many “strange” 2F1 evaluations, and these discoveries are in some sense, exhaustive. Hence WZ theory is transgressing the boundaries of the context of justification into the context of discovery.