D-Finite Functions (Algorithms and Computation in Mathematics, 30)

✍ Scribed by Manuel Kauers

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 669
• Category: Library

✦ Synopsis

Defined as solutions of linear differential or difference equations with polynomial coefficients, D-finite functions play an important role in various areas of mathematics. This book is a comprehensive introduction to the theory of these functions with a special emphasis on computer algebra algorithms for computing with them: algorithms for detecting relations from given data, for evaluating D-finite functions, for executing closure properties, for obtaining various kinds of “explicit” expressions, for factoring operators, and for definite and indefinite symbolic summation and integration are explained in detail.

The book comes “with batteries included” in the sense that it requires no background in computer algebra as the relevant facts from this area are summarized in the beginning. This makes the book accessible to a wide range of readers, from mathematics students who plan to work themselves on D-finite functions to researchers who want to apply the theory to their own work. Hundreds of exercises invite the reader to apply the techniques in the book and explore further aspects of the theory on their own. Solutions to all exercises are given in the appendix.

When algorithms for D-finite functions came up in the early 1990s, computer proofs were met with a certain skepticism. Fortunately, these times are over and computer algebra has become a standard tool for many mathematicians. Yet, this powerful machinery is still not as widely known as it deserves. This book helps to spread the word that certain tasks can be safely delegated to a computer algebra system, and also what the limitations of these techniques are.

✦ Table of Contents

Preface
Contents
1 Background and Fundamental Concepts
1.1 Functions, Sequences, and Series
Exercises
References
1.2 D-Finiteness
Exercises
References
1.3 Applications
1.4 Computer Algebra
Exercises
References
1.5 Guessing
Exercises
References
1.6 Hermite-Padé Approximation
Exercises
References
2 The Recurrence Case in One Variable
2.1 Evaluation
Exercises
References
2.2 The Solution Space
Exercises
References
2.3 Closure Properties
Exercises
References
2.4 Generalized Series Solutions
Exercises
References
2.5 Polynomial and Rational Solutions
Exercises
References
2.6 Hypergeometric and d'Alembertian Solutions
Exercises
References
3 The Differential Case in One Variable
3.1 Evaluation
Exercises
References
3.2 The Solution Space
Exercises
References
3.3 Closure Properties
Exercises
References
3.4 Generalized Series Solutions
Exercises
References
3.5 Polynomial and Rational Solutions
Exercises
References
3.6 Hyperexponential and d'Alembertian Solutions
Exercises
References
4 Operators
4.1 Ore Algebras and Ore Actions
Exercises
References
4.2 Common Right Divisors and Left Multiples
Exercises
References
4.3 Several Functions
Exercises
References
4.4 Factorization
Exercises
References
4.5 Several Variables
Exercises
References
4.6 Gröbner Bases
Exercises
References
5 Summation and Integration
5.1 The Indefinite Problem
Exercises
References
5.2 The Definite Problem
Exercises
References
5.3 Further Closure Properties
Exercises
References
5.4 Creative Telescoping
Exercises
References
5.5 Bounds
Exercises
References
5.6 Reduction-Based Algorithms
Exercises
References
Answers to Exercises
Section 1.1
Section 1.2
Section 1.4
Section 1.5
Section 1.6
Section 2.1
Section 2.2
Section 2.3
Section 2.4
Section 2.5
Section 2.6
Section 3.1
Section 3.2
Section 3.3
Section 3.4
Section 3.5
Section 3.6
Section 4.1
Section 4.2
Section 4.3
Section 4.4
Section 4.5
Section 4.6
Section 5.1
Section 5.2
Section 5.3
Section 5.4
Section 5.5
Section 5.6
Software
Mathematica
Sage
Maple
Notation
References
Index