By Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger

ISBN-10: 1568810636

ISBN-13: 9781568810638

This booklet is of curiosity to mathematicians and laptop scientists operating in finite arithmetic and combinatorics. It offers a leap forward approach for studying advanced summations. superbly written, the booklet comprises functional purposes in addition to conceptual advancements that would have purposes 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 comprises an creation to the topic and ends with a collection of routines.

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This concise, self-contained textbook supplies an in-depth examine problem-solving from a mathematician’s point-of-view. each one bankruptcy builds off the former one, whereas introducing quite a few tools which may be used whilst coming near near any given challenge. inventive pondering is the main to fixing mathematical difficulties, and this booklet outlines the instruments essential to enhance the reader’s technique.

The textual content is split into twelve chapters, each one offering corresponding tricks, causes, and finalization of ideas for the issues within the given bankruptcy. For the reader’s comfort, each one workout is marked with the mandatory heritage point. This publication implements quite a few ideas that may be used to unravel mathematical difficulties in fields comparable to research, calculus, linear and multilinear algebra and combinatorics. It contains functions to mathematical physics, geometry, and different branches of arithmetic. additionally supplied in the textual content are real-life difficulties in engineering and technology.

Thinking in difficulties is meant for complex undergraduate and graduate scholars within the lecture room or as a self-study advisor. must haves contain linear algebra and analysis.

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**Example text**

D. dissertation of Sister Mary Celine Fasenmyer, in 1945. It showed how recurrences for certain polynomial sequences could be found algorithmically. ) • [Gosp78], by R. W. , is the discovery of the algorithmic solution of the problem of indefinite hypergeometric summation (see Chapter 5). Such a summation is of the form f (n) = nk=0 F (k), where F is hypergeometric. • [Zeil82], of Zeilberger, recognized that Sister Celine’s method would also be the basis for proving combinatorial identities by recurrence.

A − b − c)! 6 1 a ;z = − (1 − z)a Γ( 12 )Γ(c + 12 )Γ( 12 + a2 + 2b )Γ( 12 − a2 − a, b, c ; 1 = 1+a+b , 2c Γ( 12 + a2 )Γ( 12 + 2b )Γ( 12 − a2 + c)Γ( 12 − 2 b 2 b 2 + c) + c) π21−2c (d − 1)! (2c + d)! a, 1 − a, c ; 1 = a−d−1 a+d d, 1 + 2c − d ( 2 )! ( 2 − 1)! (c − a+d )! ( d−a−1 )! 2 2 Using the database Let’s review where we are. In this chapter we have seen how to take a sum and identify it, when possible, as a standard hypergeometric sum. We have also seen a list of many of the important hypergeometric sums that can be expressed in simple, closed form.

K, 0, n}], we find that Mathematica is very well trained indeed, since it gives LaguerreL[n, 0, 1] which means that it recognizes our sum as a Laguerre polynomial! The trick of inserting xk won’t change this behavior, so there isn’t any way to adapt this routine to the present example. ) and then ask for3 the term ratio, FactorialSimplify[t[k+1]/t[k]]. We would obtain the term ratio in the nicely factored form k−n . 4, −n ;1 . 1F1 1 3 Read in DiscreteMath‘RSolve‘ before attempting to FactorialSimplify something.

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

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