By Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger

This e-book is of curiosity to mathematicians and laptop scientists operating in finite arithmetic and combinatorics. It offers a step forward approach for studying complicated summations. superbly written, the publication includes functional purposes in addition to conceptual advancements that might have functions in different parts of mathematics.From the desk of contents: * facts Machines * Tightening the objective * The Hypergeometric Database * The 5 simple Algorithms: Sister Celine's strategy, Gosper&'s set of rules, Zeilberger's set of rules, The WZ Phenomenon, set of rules Hyper * Epilogue: An Operator Algebra point of view * The WWW websites and the software program (Maple and Mathematica) every one bankruptcy includes an creation to the topic and ends with a collection of workouts.

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6 Where we are and what happens next We have so far discussed the following two pairs, each consisting of an identity and its WZ proof certificate: k k n k n k = 2n , = 2n , n R(n, k) = k ; 2(k − n − 1) R(n, k) = − k 2 (3n + 3 − 2k) . 2(n + 1 − k)2 (2n + 1) So what we can expect from computer methods are short, even one-line, proofs of combinatorial identities, in standardized format, as well as finding the right hand side if it is unknown. Human beings might have a great deal of trouble in finding one of these proofs, but the verification procedure, as we have seen, is perfectly civilized, and involves only a medium amount of human labor.

1F1 1 3 Read in DiscreteMath‘RSolve‘ before attempting to FactorialSimplify something. 4 Software that identifies hypergeometric series 41 To finish on a positive note, we’ll ask Mathematica to identify quite a tricky sum for us, by entering Sum[(−1)ˆkBinomial[r − s − k, k]Binomial[r − 2k, n − k]/(r − n − k + 1), {k, 0, n}]. 2) which is extremely helpful. Next let’s try a session with Maple. The capability in Maple to identify a series as a p Fq [· · · ] rests with the function convert/hypergeom.

That is, we have n k k = 2n . The sum of the squares of the binomial coefficients is something simple, too: k n k = 2n . n Range convention: Please note that, throughout this book, when ranges of summation are not specified, then the sums are understood to extend over all integers, positive and negative. In the above sum, for instance, the binomial coefficient nk vanishes if k < 0 or k > n ≥ 0 (assuming n is an integer), so only finitely many terms contribute. 20 Tightening the Target But what about the sum of their cubes?