The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates

Cover......Page 1
Texts and Monographs in Symbolic\rComputation......Page 3
The Concrete Tetrahedron......Page 4
ISBN 9783709104446......Page 5
Preface......Page 6
Contents......Page 8
1.1 Selection Sort and Quicksort......Page 12
1.2 Recurrence Equations......Page 16
1.3 Symbolic Sums......Page 18
1.4 Generating Functions......Page 19
1.5 Asymptotic Estimates......Page 23
1.6 The Concrete Tetrahedron......Page 24
1.7 Problems......Page 26
2.1 Basic Facts and Definitions......Page 28
2.2 Differentiation and Division......Page 30
2.3 Sequences of Power Series......Page 35
2.4 The Transfer Principle......Page 40
2.5 Multivariate Power Series......Page 42
2.6 Truncated Power Series......Page 46
2.7 Problems......Page 49
3.1 Polynomials as Power Series......Page 54
3.2 Polynomials as Sequences......Page 57
3.3 The Tetrahedron for Polynomials......Page 59
3.4 Polynomials as Solutions......Page 61
3.5 Polynomials as Coefficients......Page 64
3.6 Applications......Page 66
3.7 Problems......Page 69
4.1 Fibonacci Numbers......Page 74
4.2 Recurrences with Constant Coefficients......Page 77
4.3 Closure Properties......Page 81
4.4 The Tetrahedron for C-finite Sequences......Page 85
4.5 Systems of C-finite Recurrences......Page 89
4.6 Applications......Page 91
4.7 Problems......Page 95
5.1 The Binomial Theorem......Page 98
5.2 Basic Facts and Definitions......Page 101
5.3 The Tetrahedron for Hypergeometric Sequences......Page 105
5.4 Indefinite Summation......Page 109
5.5 Definite Summation......Page 114
5.6 Applications......Page 118
5.7 Problems......Page 121
6.1 Catalan Numbers......Page 124
6.2 Basic Facts and Definitions......Page 127
6.3 Puiseux Series and the Newton Polygon......Page 130
6.4 Closure Properties......Page 134
6.5 The Tetrahedron for Algebraic Functions......Page 136
6.6 Applications......Page 141
6.7 Problems......Page 144
7.1 Harmonic Numbers......Page 148
7.2 Equations with Polynomial Coefficients......Page 150
7.3 Generalized Series Solutions......Page 155
7.4 Closed Form Solutions......Page 159
7.5 The Tetrahedron for Holonomic Functions......Page 163
7.6 Applications......Page 168
7.7 Problems......Page 171
A.1 Basic Notions and Notations......Page 176
A.2 Basic Facts from Computer Algebra......Page 178
A.3 A Collection of Formal Power Series Identities......Page 179
A.4 Closure Properties at One Glance......Page 180
A.5 Software......Page 182
A.6 Solutions to Selected Problems......Page 185
A.7 Bibliographic Remarks......Page 200
References......Page 204
Subject Index......Page 208