By Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger

This e-book is of curiosity to mathematicians and machine scientists operating in finite arithmetic and combinatorics. It offers a step forward process for reading complicated summations. fantastically written, the publication includes useful purposes in addition to conceptual advancements that would have purposes in different parts of mathematics.From the desk of contents: * facts Machines * Tightening the objective * The Hypergeometric Database * The 5 easy 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 standpoint * The WWW websites and the software program (Maple and Mathematica) each one bankruptcy includes an advent to the topic and ends with a suite of workouts.

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**Example text**

2n corresponds uniquely to such a pair of choices, for some k. ✷ We must pause to remark that that one is a really nice proof. So as we go through this book whose main theme is that computers can prove all of these identities, please note that we will never7 claim that computerized proofs are better than human ones, in any sense. When an elegant proof exists, as in the above example, the computer will be hard put to top it. On the other hand, the contest will be close even here, because the computerized proof that’s coming up is rather elegant too, in a different way.

4, −n ;1 . 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.

So we must be doing something wrong. Well, it turns out that if you would really like to simplify ratios of factorials then the thing to do is to read in the package RSolve, because in that package there lives a command FactorialSimplify, which does the simplification that you would like to see. Out[2] = 28 Tightening the Target So let’s start over, this time with In[1] :=<< DiscreteMath‘RSolve‘. ] and we get Out[2] = 1 + n, which is what we wanted. 1). ). The rational function certificate (the key to the lock) for this identity is R(n, k) = − k 2 (3n + 3 − 2k) .