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

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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.

A database D is a triple consisting of 1. a collection of information (data) and 2. a collection of queries (questions) that may be addressed to the database D by the user and 3. a collection of algorithms by which the system responds to queries and searches the data in order to find the answers. There is no hypergeometric database. It’s a myth. Is there a collection of information? The data might be, for instance, a list of all known hypergeometric identities. But there isn’t any such list. If you propose one, somebody will produce a known identity that isn’t on your list.

1) The first step is, as always, to find the term ratio and resolve it into linear factors. We find that 2n (−1)k+1 k+1 tk+1 = tk (−1)k 2n k 2k+2 4n−2k−2 k+1 2n−k−1 2k 4n−2k k 2n−k (k + 12 )(k − 2n)2 = . (k + 1)2(k − 2n + 12 ) The first term of our sum is not 1, but is instead desired sum is f(n) = 4n 2n 3F2 4n 2n −2n −2n 1 −2n + 12 , hence we now know that our 1 2 ;1 . 2) Since this is a 3 F2, we check the possibilities of Saalsch¨ utz’s identity and Dixon’s identity. It does not match the condition d + e = a + b + c + 1 of Saalsch¨ utz, so we 1 try Dixon’s next.

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A=B by Petkovsek M., Wilf H.S., Zeilberger D.

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