The Concrete Tetrahedron

by Kauers, Manuel; Paule, Peter

• ISBN13: 9783709104446
• ISBN10: 3709104440
• Format: Paperback
• Copyright: 2011-02-11
• Publisher: Springer Verlag

Summary

The book treats four mathematical concepts which play a great role in many different areas of mathematics: symbolic sums, recurrence (difference) equations, generating functions, and asymptotic estimates. Their key features in isolation or in combination, their mastery by paper and pencil or by computer programs, and their applications to problems in pure mathematics or to "real world problems" are studied. The book can be viewed as an algorithmic version of the bestselling "Concrete Mathematics" by Graham, Knuth and Patashnik.

Table of Contents

Introduction	p. 1
Selection Sort and Quicksort	p. 1
Recurrence Equations	p. 5
Symbolic Sums	p. 7
Generating Functions	p. 8
Asymptotic Estimates	p. 12
The Concrete Tetrahedron	p. 13
Problems	p. 15
Formal Power Series	p. 17
Basic Facts and Definitions	p. 17
Differentiation and Division	p. 19
Sequences of Power Series	p. 24
The Transfer Principle	p. 29
Multivariate Power Series	p. 31
Truncated Power Series	p. 35
Problems	p. 38
Polynomials	p. 43
Polynomials as Power Series	p. 43
Polynomials as Sequences	p. 46
The Tetrahedron for Polynomials	p. 48
Polynomials as Solutions	p. 50
Polynomials as Coefficients	p. 53
Applications	p. 55
Problems	p. 58
C-Finite Sequences	p. 63
Fibonacci Numbers	p. 63
Recurrences with Constant Coefficients	p. 66
Closure Properties	p. 70
The Tetrahedron for C-finite Sequences	p. 74
Systems of C-finite Recurrences	p. 78
Applications	p. 80
Problems	p. 84
Hypergeometric Series	p. 87
The Binomial Theorem	p. 87
Basic Facts and Definitions	p. 90
The Tetrahedron for Hypergeometric Sequences	p. 94
Indefinite Summation	p. 98
Definite Summation	p. 103
Applications	p. 107
Problems	p. 110
Algebraic Functions	p. 113
Catalan Numbers	p. 113
Basic Facts and Definitions	p. 116
Puiseux Series and the Newton Polygon	p. 119
Closure Properties	p. 123
The Tetrahedron for Algebraic Functions	p. 125
Applications	p. 130
Problems	p. 133
Holonomic Sequences and Power Series	p. 137
Harmonic Numbers	p. 137
Equations with Polynomial Coefficients	p. 139
Generalized Series Solutions	p. 144
Closed Form Solutions	p. 148
The Tetrahedron for Holonomic Functions	p. 152
Applications	p. 157
Problems	p. 160
Appendix	p. 165
Basic Notions and Notations	p. 165
Basic Facts from Computer Algebra	p. 167
A Collection of Formal Power Series Identities	p. 168
Closure Properties at One Glance	p. 169
Software	p. 171
Solutions to Selected Problems	p. 174
Bibliographic Remarks	p. 189
References	p. 193
Subject Index	p. 197
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