By Mohamud Mohammed and Doron Zeilberger
First Version Written: June 3, 2004, this much Expanded Version Written: Aug. 10, 2004.
Zeilberger's algorithm starts out with ORDER=0 and climbs up until success is reached. How do we know that it halts? If we can't prove it, then it is not an algorithm. The first proof of termination relied on the deep theory of D-modules, pioneered by Joseph Bernstein, and no a priori upper bound was given, only the fact that there exists a successful ORDER. Then W and Z found an explicit upper bound, using Sister Celine's method. But their `explicit' upper bound was very dull, as countless experimentations with Zeil in EKHAD showed. In this article, Mohamud Mohammed and I show that the Z-algorithm is self-sufficient, i.e. it can use itself to find sharp upper bounds for the ORDER. But, for this meta-application of the Z-algorithm we still need humans (at least for now, I won't be surprised if soon this article would be written by computers), since the input is no longer a specific proper-hypergeometric function, but rather the generic form.