This page intentionally left blank [50] Develop computer programs for simplifying sums that involve binomial coe±cients. Exercise 1.2.6.63 in The Art of Computer Programming, Volume 1: Fundamental Algorithms by Donald E. Knuth, Addison Wesley, Reading, Massachusetts, 1968. A=B Marko Petkov·sek Herbert S. Wilf University of Ljubljana University of Pennsylvania Ljubljana, Slovenia Philadelphia, PA, USA Doron Zeilberger Temple University Philadelphia, PA, USA April 27, 1997 ii Contents Foreword vii AQuickStart::: ix I Background 1 1 Proof Machines 3 1.1Evolutionoftheprovinceofhumanthought.............. 3 1.2Canonicalandnormalforms....................... 7 1.3Polynomialidentities........................... 8 1.4Proofsbyexample?............................ 9 1.5Trigonometricidentities......................... 11 1.6Fibonacciidentities............................ 12 1.7Symmetricfunctionidentities...................... 12 1.8 Elliptic function identities ........................ 13 2 Tightening the Target 17 2.1Introduction................................ 17 2.2Identities.................................. 21 2.3Humanandcomputerproofs;anexample................ 23 2.4AMathematicasession.......................... 27 2.5AMaplesession.............................. 29 2.6Whereweareandwhathappensnext.................. 30 2.7Exercises.................................. 31 3 The Hypergeometric Database 33 3.1Introduction................................ 33 3.2Hypergeometricseries........................... 34 3.3Howtoidentifyaseriesashypergeometric............... 35 3.4Softwarethatidenti¯eshypergeometricseries.............. 39 iv CONTENTS 3.5Someentriesinthehypergeometricdatabase.............. 42 3.6Usingthedatabase............................ 44 3.7Istherereallyahypergeometricdatabase?............... 48 3.8Exercises.................................. 50 II The Five Basic Algorithms 53 4 Sister Celine's Method 55 4.1Introduction................................ 55 4.2SisterMaryCelineFasenmyer...................... 57 4.3SisterCeline'sgeneralalgorithm..................... 58 4.4 The Fundamental Theorem ....................... 64 4.5 Multivariate and \q"generalizations.................. 70 4.6Exercises.................................. 72 5 Gosper's Algorithm 73 5.1Introduction................................ 73 5.2Hypergeometricstorationalstopolynomials.............. 75 5.3Thefullalgorithm:Step2........................ 79 5.4Thefullalgorithm:Step3........................ 84 5.5Moreexamples.............................. 86 5.6Similarityamonghypergeometricterms................. 91 5.7Exercises.................................. 95 6 Zeilberger's Algorithm 101 6.1Introduction................................ 101 6.2Existenceofthetelescopedrecurrence.................. 104 6.3Howthealgorithmworks......................... 106 6.4Examples................................. 109 6.5Useoftheprograms........................... 112 6.6Exercises.................................. 118 7 The WZ Phenomenon 121 7.1Introduction................................ 121 7.2WZproofsofthehypergeometricdatabase............... 126 7.3Spino®sfromtheWZmethod...................... 127 7.4Discoveringnewhypergeometricidentities............... 135 7.5SoftwarefortheWZmethod....................... 137 7.6Exercises.................................. 140 CONTENTS v 8 Algorithm Hyper 141 8.1Introduction................................ 141 8.2Theringofsequences........................... 144 8.3Polynomialsolutions........................... 148 8.4Hypergeometricsolutions......................... 151 8.5AMathematicasession.......................... 156 8.6Findingallhypergeometricsolutions.................. 157 8.7Findingallclosedformsolutions..................... 158 8.8Somefamoussequencesthatdonothaveclosedform......... 159 8.9Inhomogeneousrecurrences........................ 161 8.10Factorizationofoperators........................ 162 8.11Exercises.................................. 164 III Epilogue 169 9 An Operator Algebra Viewpoint 171 9.1Earlyhistory............................... 171 9.2Lineardi®erenceoperators........................ 172 9.3Eliminationintwovariables....................... 177 9.4Modi¯edeliminationproblem...................... 180 9.5Discreteholonomicfunctions....................... 184 9.6Eliminationintheringofoperators................... 185 9.7Beyondtheholonomicparadigm..................... 185 9.8Bi-basicequations............................. 187 9.9Creativeanti-symmetrizing........................ 188 9.10Wavelets.................................. 190 9.11Abel-typeidentities............................ 191 9.12Anothersemi-holonomicidentity.................... 193 9.13Theart.................................. 193 9.14Exercises.................................. 195 A The WWW sites and the software 197 A.1 The Maple packages EKHAD and qEKHAD ................. 198 A.2Mathematicaprograms.......................... 199 Bibliography 201 Index 208 vi CONTENTS Foreword Science is what we understand well enough to explain to a computer. Art is everything else we do. During the past several years an important part of mathematics has been transformed from an Art to a Science: No longer do we need to get a brilliant insight in order to evaluate sums of binomial coe±cients, and many similar formulas that arise frequently in practice; we can now follow a mechanical procedure and discover the answers quite systematically. I fell in love with these procedures as soon as I learned them, because they worked for me immediately. Not only did they dispose of sums that I had wrestled with long and hard in the past, they also knocked o® two new problems that I was working on at the time I ¯rst tried them. The success rate was astonishing. In fact, like a child with a new toy, I can't resist mentioning how I used the new P ³ ´³ ´ 2n¡2k 2k methods just yesterday. Long ago I had run into the sum k n¡k k ,whichtakes the values 1, 4, 16, 64 for n =0,1,2,3soitmustbe4n. Eventually I learned a tricky way to prove that it is, indeed, 4n; but if I had known the methods in this book I could have proved the identity immediately. Yesterday I was working on a harder problem P ³ ´ ³ ´ 2n¡2k 2 2k 2 whose answer was Sn = k n¡k k . I didn't recognize any pattern in the ¯rst values 1, 8, 88, 1088, so I computed away with the Gosper-Zeilberger algorithm. In 3 ¡ 1 2 ¡ ¡ ¡ 3 afewminutesIlearnedthatn Sn =16(n 2 )(2n 2n +1)Sn¡1 256(n 1) Sn¡2. Notice that the algorithm doesn't just verify a conjectured identity \A = B". It also answers the question \What is A?", when we haven't been able to formulate a decent conjecture. The answer in the example just considered is a nonobvious recurrence from which it is possible to rule out any simple form for Sn. I'm especially pleased to see the appearance of this book, because its authors have not only played key roles in the new developments, they are also master expositors of mathematics. It is always a treat to read their publications, especially when they are discussing really important stu®. Science advances whenever an Art becomes a Science. And the state of the Art ad- vances too, because people always leap into new territory once they have understood more about the old. This book will help you reach new frontiers. Donald E. Knuth Stanford University 20 May 1995 viii CONTENTS A Quick Start ::: You've been up all night working on your new theory, you found the answer, and it's in the form of a sum that involves factorials, binomial coe±cients, and so on, such as à !à ! Xn ¡ ¡ ¡ k x k +1 x 2k f(n)= ( 1) ¡ : k=0 k n k You know that many sums like this one have simple evaluations and you would like to know, quite de¯nitively, if this one does, or does not. Here's what to do. 1. Let F (n; k)beyoursummand, i.e., the function1 that is being summed. Your ¯rst task is to ¯nd the recurrence that F satis¯es. 2. If you are using Mathematica, go to step 4 below. If you are using Maple, then get the package EKHAD either from the included diskette or from the World- WideWeb site given on page 197. Read in EKHAD,andtype zeil(F(n; k); k; n; N); in which your summand is typed, as an expression, in place of \F(n,k)". So in the example above you might type f:=(n,k)->(-1)^k*binomial(x-k+1,k)*binomial(x-2*k,n-k); zeil(f(n,k),k,n,N); Then zeil will print out the recurrence that your summand satis¯es (it does satisfy one; see theorems 4.4.1 on page 65 and 6.2.1 on page 105). The output recurrence will look like eq. (6.1.3) on page 102. In this example zeil prints out the recurrence ((n +2)(n ¡ x) ¡ (n +2)(n ¡ x)N 2)F(n; k)=G(n; k +1)¡ G(n; k); 1But what is the little icon in the right margin? See page 9. x A Quick Start ::: where N is the forward shift operator and G is a certain function that we will ignore for the moment. In customary mathematical notation, zeil will have found that (n +2)(n ¡ x)F(n; k) ¡ (n +2)(n ¡ x)F (n +2;k)=G(n; k +1)¡ G(n; k): 3. The next step is to sum the recurrence that you just found over all the values of k that interest you. In this case you can sum over all integers k.Theright side telescopes to zero, and you end up with the recurrence that your unknown sum f(n) satis¯es, in the form f(n) ¡ f(n +2)=0: Since f(0) = 1 and f(1) = 0, you have found that f(n)=1,ifn is even, and f(n)=0,ifn is odd, and you're all ¯nished. If, on the other hand, you get a recurrence whose solution is not obvious to you because it is of order higher than the ¯rst and it does not have constant coe±cients, for instance, then go to step 5 below. 4. If you are using Mathematica, then get