The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates / Edition 1

The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates / Edition 1

by Manuel Kauers, Peter Paule
     
 

ISBN-10: 3709104440

ISBN-13: 9783709104446

Pub. Date: 01/19/2011

Publisher: Springer Vienna

The book treats four mathematical concepts which play a fundamental 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

Overview

The book treats four mathematical concepts which play a fundamental 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" (e.g. the analysis of algorithms) are studied. The book is intended as an algorithmic supplement to the bestselling "Concrete Mathematics" by Graham, Knuth and Patashnik.

Product Details

ISBN-13:
9783709104446
Publisher:
Springer Vienna
Publication date:
01/19/2011
Series:
Texts & Monographs in Symbolic Computation Series
Edition description:
2011
Pages:
214
Product dimensions:
6.00(w) x 9.20(h) x 0.40(d)

Table of Contents

1 Introduction

1.1 Selection Sort and Quicksort

1.2 Recurrence Equations

1.3 Symbolic Sums

1.4 Generating Functions

1.5 Asymptotic Estimates

1.6 The Concrete Tetrahedron

1.7 Problems

2 Formal Power Series

2.1 Basic Facts and Definitions

2.2 Differentiation and Division

2.3 Sequences of Power Series

2.4 The Transfer Principle

2.5 Multivariate Power Series

2.6 Truncated Power Series

2.7 Problems

3 Polynomials

3.1 Polynomials as Power Series

3.2 Polynomials as Sequences

3.3 The Tetrahedron for Polynomials

3.4 Polynomials as Solutions

3.5 Polynomials as Coefficients

3.6 Applications

3.7 Problems

4 C-Finite Sequences

4.1 Fibonacci Numbers

4.2 Recurrences with Constant Coefficients

4.3 Closure Properties

4.4 The Tetrahedron for C-finite Sequences

4.5 Systems of C-finite Recurrences

4.6 Applications

4.7 Problems

5 Hypergeometric Series

5.1 The Binomial Theorem

5.2 Basic Facts and Definitions

5.3 The Tetrahedron for Hypergeometric Sequences

5.4 Indefinite Summation

5.5 Definite Summation

5.6 Applications

5.7 Problems

6 Algebraic Functions

6.1 Catalan Numbers

6.2 Basic Facts and Definitions

6.3 Puiseux Series and the Newton Polygon

6.4 Closure Properties

6.5 The Tetrahedron for Algebraic Functions

6.6 Applications

6.7 Problems

7 Holonomic Sequences and Power Series

7.1 Harmonic Numbers

7.2 Equations with Polynomial Coefficients

7.3 Generalized Series Solutions

7.4 Closed Form Solutions

7.5 The Tetrahedron for Holonomic Functions

7.6 Applications

7.7 Problems

Appendix

A.1 Basic Notions and Notations

A.2 Basic Facts from Computer Algebra

A.3 A Collection of Formal Power Series Identities

A.4 Closure Properties at One Glance

A.5 Software

A.6 Solutions to Selected Problems

A.7 Bibliographic Remarks

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