By Petkovsek M., Wilf H.S., Zeilberger D.

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K! (n − k)! 2k! (n − k)! 2(k − 1)! (n − k + 1)! ” To clear out all of the factorials we multiply through by k! , and get n+1 n+1−k k − (n + 1 − k) = − + , 2 2 2 which is really trivial. 4 A Mathematica session 27 In Step 5 of the standardized WZ algorithm we must check that But 1, if k = 0; 0 F (0, k) = = 0, otherwise, k k F (0, k) = 1. and we’re all finished (that’s also the last time we’ll do a routine verification in full). 2. In an article in the American Mathematical Monthly 101 (1994), p.

That is, if they produce nothing, then that is a proof that 49 50 The Hypergeometric Database nothing exists, rather than only a confession of possible inadequacy. Obviously we cannot claim that the computerized methods are the best for every situation. Sometimes the certificates that they produce are longer and less userfriendly than those that humans might find, for example. But the emergence of these methods has put an important family of tools in the hands of discrete mathematicians, and many results that are accessible in no other way have been found and proved by computer methods.

This book contains both mathematics and software, the former being the theoretical underpinnings of the latter. For those who have not previously used them, the programs will likely be a revelation. Imagine the convenience of being able to input a sum that one is interested in and having the program print out a simple formula that evaluates it! Think also of inputting a complicated sum and getting a recurrence formula that it satisfies, automatically. We hope you’ll enjoy both the mathematics and the software.

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