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This book examines four mathematical concepts which play a great role in many different areas of mathematics: symbolic sums, recurrence (difference) equations, generating functions, and asymptotic estimates.

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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.

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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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From the reviews:

"The Concrete Tetrahedron is suitable for advanced undergraduate students in mathematics or computer science who have had some exposure to elementary concepts in abstract algebra and complex analysis. ... The volume´s numerous problems together with selected solutions definitely make it appropriate for an independent reading course for the motivated student. Summing Up: Highly recommended. Upper-division undergraduates." (D. M. Ha, Choice, Vol. 49 (11), July, 2012)

"The book deals with a blend of continuous and discrete mathematics, applying continuous methods to discrete problems. ... Those who are interested in the subject matter of Concrete Mathematics will enjoy The Concrete Tetrahedron for an independent perspective, some more recent results, and for new applications." (John D. Cook, The Mathematical Association of America, March, 2011)

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