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The first two papers in this series raise the following obvious question: Why should anyone want to resurrect a method last used in 1797 to sum hypergeometric series when it is well-known that the WZ method [13] has swept all before it.

This latter state of affairs has been spelled out in delightful albeit
idiosyncratic detail by Zeilberger in
[16] and [17]. The reader
is urged to consult these references for the complete understanding of
his philosophy. Perhaps the case can be put succinctly by referring
to his Meta-theorem [16] which asserts that the verification of
* any* binomial coefficient identity via the WZ method is routine.

To be fair, it should be noted that Zeilberger softens this position a bit in [17]:

``This algorithm [the WZ method] can be performed successfully on all natural identities we are now aware of. It is easy, however, to concoct artificial examples for which the running time and memory are prohibitive. Undoubtedly, in the future natural identities will be encountered whose complete proof will turn out to be not worth the money.''To my mind, Zeilberger's mathematical discoveries on this topic are powerful and important while his philosophical conclusions are quite wrong-headed.

It is certainly true that the WZ method has been effectively automated by Zeilberger [14]; however, the computational complexity of the method has not been carefully examined to my knowledge. This state of affairs will become much clearer (or perhaps less clear) in Section 5 where we discuss a theorem (eq. (5.5)), that may well not be worth the money to prove via the WZ method. While I hope that some of the grandiose claims in [16] and [17] for the WZ method might be modified, the last thing I wish to suggest is that Pfaff's method is a replacement. I shall emphasize my admiration for the WZ method in Section 7.

Section 2--4 will compare proofs of theorems for which both Pfaff and WZ are quite successful.

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