By Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger

This e-book is of curiosity to mathematicians and desktop scientists operating in finite arithmetic and combinatorics. It offers a step forward strategy for examining advanced summations. fantastically written, the ebook comprises sensible purposes in addition to conceptual advancements that would have purposes in different components of mathematics.From the desk of contents: * evidence Machines * Tightening the objective * The Hypergeometric Database * The 5 uncomplicated Algorithms: Sister Celine's process, Gosper&'s set of rules, Zeilberger's set of rules, The WZ Phenomenon, set of rules Hyper * Epilogue: An Operator Algebra perspective * The WWW websites and the software program (Maple and Mathematica) every one bankruptcy comprises an advent to the topic and ends with a suite of workouts.

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**Sample text**

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. But suppose that problem didn’t exist. Let’s compromise a bit, and settle for a very large collection of many of the most important hypergeometric identities. Fine. Now what are the queries that we would like to address to the database? That’s a lot easier. ” All right.

1) 2j , j = 0≤j≤n tm m! 2m s 2n , n (n + 1)n−1 =1+ n≥1 = (−1)m (3m)! 2) tn , n! 4) Beautiful identities have often stimulated mathematicians to find correspondingly beautiful proofs for them; proofs that have perhaps illuminated the combinatorial or other significance of the equality of the two members, or possibly just dazzled us with their unexpected compactness and elegance. It is a fun activity for people to try to prove identities. We have been accused of taking the fun out of it by developing these computer methods6 but we hope that we have in fact only moved the fun to a different level.

Fine. Now what are the queries that we would like to address to the database? That’s a lot easier. ” All right. We’re trying to construct a collection of identities that will be equipped to discover if somebody’s sum can or cannot be expressed in a much simpler form. We’re two-thirds of the way there. We have a (slightly mythical) collection of data, and a single rather precise query. What we are missing is the algorithm. If some user asks the system whether or not a certain sum can be expressed in some simple form, exactly what steps shall the system take in order to answer the question?