By Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger

This publication is of curiosity to mathematicians and laptop scientists operating in finite arithmetic and combinatorics. It offers a step forward strategy for studying advanced summations. superbly written, the ebook comprises sensible functions in addition to conceptual advancements that may have functions in different components of mathematics.From the desk of contents: * evidence Machines * Tightening the objective * The Hypergeometric Database * The 5 easy Algorithms: Sister Celine's procedure, 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) each one bankruptcy includes an advent to the topic and ends with a collection of routines.

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Hypergeometric series are very important in mathematics. Many of the familiar functions of analysis are hypergeometric. These include the exponential, logarithmic, trigonometric, binomial, and Bessel functions, along with the classical orthogonal polynomial sequences of Legendre, Chebyshev, Laguerre, Hermite, etc. 3 How to identify a series as hypergeometric 35 metric, then identifying precisely which hypergeometric function it is, and finally by using known results about such functions. 1) where x is a constant.

Define e(x) to be the famous series n≥0 xn /n!. We will prove that e(x + y) = e(x)e(y) for all x and y. First, the series converges for all x, by the ratio test, so e(x) is well defined for all x, and e (x) = e(x). Next, instead of trying to prove that the two sides of the identity are equal, let’s prove that their ratio is 1 (that will be a frequent tactic in 1 See page 143. 2 Identities 21 this book). Not only that, we’ll prove that the ratio is 1 by differentiating it and getting 0 (another common tactic here).

2n. Hence there are nk n−k = nk ways to make ways of choosing n letters from the such a pair of choices. But every one of the 2n n 2n letters 1, 2, . . , 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.