By Wilf, Zeilberger.

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Here we will not, of course, be able to discuss all kinds of identities. Far from it. 4) above. The main purpose of this book is to explain how the discoveries and the proofs of hypergeometric identities have been very largely automated. The book is not primarily about computing; it is the mathematics that underlies the computing that will be the main focus. Automating the discovery and proof of identities is not something that is immediately obvious as soon as you have a large computer. The theoretical developments that have led to the automation make what we believe is a very interesting story, and we would like to tell it to you.

2n − 12 )! 2 2n . n (2n − 12 )! (− 12 )! But for every positive integer m, 1 1 3 1 1 (m − )! = (m − )(m − ) · · · ( )(− )! 2 2 2 2 2 (2m − 1)(2m − 3) · · · 1 1 = (− )! 2m 2 (2m)! 1 = m (− )!. 4 m! 6 Using the database 47 So we can simplify our answer all the way down to f(n) = labor is that we have found the identity (−1)k k 2n k 2k k 2n 2 . 3) we realize that it is a special case of Dixon’s identity, and we further realize that the “lookup” in the database was not quite a routine matter! ✷ Since that was a very tedious lookup operation, might one of our computer packages have been able to help?

Bq k≥0 (b1 )k (b2 )k · · · (bq )k k! 2) The series is well defined as long as the lower parameters b1 , b2 , . . , bq are not negative integers or zero. The series terminates automatically if any of the upper parameters a1 , a2 , . . , it is an infinite series. If the series is well defined and nonterminating, then questions of convergence or divergence become relevant. In this book we will be concerned for the most part with terminating series. 4 Software that identifies hypergeometric series The act of taking a series and finding out exactly which p Fq [.