By Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger

This ebook is of curiosity to mathematicians and laptop scientists operating in finite arithmetic and combinatorics. It offers a step forward strategy for interpreting advanced summations. superbly written, the booklet comprises useful functions in addition to conceptual advancements that would have functions 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 technique, 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 creation to the topic and ends with a suite of routines.

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2) above. Finally we illustrate the use of the package Hyp. at, at the University of Vienna, in Austria. It is written in Mathematica source code and must be used in conjunction with Mathematica. To use it to identify a hypergeometric series involves the following steps. First enter the sum that interests you using the usual Sum construct. Give the expression a name, say mysum. SumF, and you will, or should, be looking at the hypergeometric designation of your sum as output. 42 The Hypergeometric Database As an example, take the Laguerre polynomial that we tried in Mathematica.

5 Some entries in the hypergeometric database The hypergeometric database can be thought of as the collection of all known hypergeometric identities. The following are some of the most useful database entries. We will not prove any of them just now because all of their proofs will follow instantly from the computer certification methods that we will develop in Chapters 4–7. On the right hand sides of these identities you will find any of three different widely used notations: rising factorial, factorial, and gamma function.

Roughly (see the formal definition below), a hypergeometric sum is one in which the summand involves only factorials, polynomials, and exponential functions of the summation variable. 1 The fact is that many hypergeometric sums can be expressed in simple closed form, and many others can be revealed to be equal to some other, seemingly different, hypergeometric sum. Whenever this happens we have an identity. , and has on the right a considerably simpler function that is equal to the sum on the left.