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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.
List of contents
Chapter. 1. Background and fundamental concepts.- Chapter. 2. The recurrence case in one variable.- Chapter. 3. The differential case in one variable.- Chapter. 4. Operators.- Chapter. 5. Summation and integration.- Answers to Exercises.- Software.- Notations.- References.- Index.
About the author
Manuel Kauers studied computer science in Karlsruhe, Germany, from 1998 to 2002 and then went to Linz, Austria, where he received his Ph.D. in symbolic computation in 2005. He won a START prize in 2009. Since 2015 he is director of the Institute for Algebra at Johannes Kepler University in Linz. Kauers is an active member of the computer algebra community and has been contributing to the design, implementation, and application of algorithms for D-finite functions for many years. Together with Christoph Koutschan and Doron Zeilberger, he proved two outstanding conjectures in enumerative combinatorics using such algorithms. For one of these results, the proof of the qTSPP-conjecture, they received the AMS David P. Robbins prize in 2016.
Report
The aim of this book is to provide a solid foundation for researchers who wish to enter the field of D-finite functions. It offers a comprehensive and self-contained introduction to the theory, computation, and applications of D-finite functions, making it particularly valuable for researchers and graduate students. The book contains many illustrative examples to facilitate the reading of the book. (Amir Hashemi, zbMATH 07756344, 2024)
The book is written in a precise style with attention to detail. The provided algorithms are in symbolic code, but can be easily (relatively) translated to programs. The exercises are well designed and informative. The book is an excellent source of information and will work best for graduate-level math education, though it may also motivate researchers to apply the theory in their own fields. (Alexander Tzanov, Computing Reviews, March 27, 2024)
