By Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger

ISBN-10: 1568810636

ISBN-13: 9781568810638

This ebook is of curiosity to mathematicians and machine scientists operating in finite arithmetic and combinatorics. It provides a step forward procedure for reading complicated summations. superbly written, the booklet includes useful functions in addition to conceptual advancements that might have purposes in different components of mathematics.From the desk of contents: * facts Machines * Tightening the objective * The Hypergeometric Database * The 5 easy Algorithms: Sister Celine's approach, 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) every one bankruptcy comprises an advent to the topic and ends with a suite of routines.

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Would you like to know how all of that is done? Read on. 3 The sum k nk , of course, is just one of many examples of formulas that can be treated with these methods. If you aren’t interested in finding or proving an identity, you might well be interested in finding a recurrence relation that an unknown sum satisfies. Or in deciding whether a given linear recurrence relation with polynomial coefficients can be solved in some explicit way. In that case this book has some powerful tools for you to use.

We take our rational function R(n, k) = k/(2(k −n−1)), and we define a new function k n −n 2 2(k − n − 1) k n kn! 2−n =− 2−n−1 . =− 2(n + 1 − k)k! (n − k)! k−1 G(n, k) = R(n, k)F (n, k) = Step 4 informs us that we will have now the equation F (n + 1, k) − F (n, k) = G(n, k + 1) − G(n, k). Let’s see if that is so. In other words, is it true that n + 1 −n−1 n −n n −n−1 n 2 − 2 =− 2 + 2−n−1 ? k k k k−1 Well, at this point we have arrived at a situation that will be referred to throughout this book as a “routinely verifiable” identity.

1 = 3F2 d|c| (d − a − b)|c| d e (IV) Dixon’s identity. In prettier and easier-to-remember form this identity reads as a+b a+c b+c (a + b + c)! (−1)k = a! b! c! a + k c + k b + k k Translated into formal hypergeometric language, it becomes the statement that, if 1 + a2 − b − c has positive real part, and if d = a − b + 1 and e = a − c + 1, then ( a2 )! (a − b)! (a − c)! ( a2 − b − c)! a b c . ;1 = 3F2 d e a! ( a2 − b)! ( a2 − c)! (a − b − c)! (V) Clausen’s 4 F3 identity. If d is a nonpositive integer and a + b + c − d = 12 , and e = a + b + 12 , and a + f = d + 1 = b + g, then 4F3 (2a)|d| (a + b)|d| (2b)|d| a b c d .

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A=B by Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger

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