Every mathematician, and user of mathematics, needs to manipulate sums or to find and handle combinatorial identities. In this book, the author provides a coherent tour of many known finite algebraic sums and offers a guide for devising simple ways of changing a given sum to a standard form that can be evaluated . As such, Summa Summarum serves as both an introduction and a reference for researchers, graduate and upper-level undergraduate students, and non-specialists alike: from tools as distinct as the most classical ideas of Euler to the recent effective computer algorithms by Gosper and Wilf-Zeilberger. The book is self-contained with relatively few prerequisites and so should be accessible to a very broad readership.
This represents the first in the new Canadian Mathematical Society Treatises in Mathematics series of books: a collection of short monographs, dedicated to well defined subjects of current interest. These treatises emphasize the interdisciplinary character of the mathematical sciences and facilitate integration of methods and results from different areas of current research.
Table of Contents
1 Notation, 2 Elementary Properties, 3 Polynomials, 4 Linear Difference Equations, 5 Classification of Sums, 6 Gosper’s Algorithm, 7 Sums of Type II(1,1,z), 8 Sums of Type II(2,2,z), 9 Sums of Type II(3,3,z), 10 Sums of Type II(4,4,±1), 11 Sums of Type II(5,5,1), 12 Other Type II Sums, 13 Zeilberger’s Algorithm, 14 Sums of Types III–IV, 15 Sums of Type V, Harmonic Sums, A Indefinite Sums, B Basic Identities
Larsen \, Mogens Esrom
" ""Introducing a wealth of useful tools from the classical concepts of Euler to modern computer algorithms discovered by Gosper as well as Wilf and Zeilberger, Summa Summarum is a superb reference and study text."" -Midwest Book Review, October 2007
""Larsens Buch liefert ein lange vermisstes Desiderat mit Bravour. Es wird sich zweifellos in kurzer Zeit zu einem Standardwerk entwickeln und darf auf keinem Buecherregal fehlen, wo es in Reichweite bleiben sollte. Die naechste Summe kommt bestimmt!"" ""Larsen’s book provides a long-awaited desideratum, in great style. Without a doubt, in a short period of time the book will be a standard work and must not be missing from any book shelf, where it should be within grasp. The next summation will come up for certain!” -Mathematische Semesterberichte, February 2008
""This is a title in a new series of monographs from the Canadian Mathematics Society and A K Peters, Ltd. introducing subjects of current interest."" -SciTech Book News, March 2008
""As both a reference and an introduction to the art of manipulating sums for graduate and upper-level undergraduate students, researchers, and non-specialists, this book provides an array of systematic techniques that will help the reader to evaluate almost any finite algebraic sum."" -L'Enseignement Mathématique, July 2007
textbook attempts to collect known algebraic finite sums in a concise and accessible way. A standardized notation is developed to make a unique representation of summations. The use of binomial coefficients and hypergeometric series notation is avoided whenever possible. Techniques are also covered that the reader may need to tackle a new summation, e.g. Gosper's algorithm or the Zeilberger's algorithm. An appendix contains basic generalizations of the simplest combinatorial identities. Anyone who had to evaluate summations will find this book useful. -Laszlo A. Szekely, Zentralblatt MATH, March 2008
The book is quite comprehensive and discusses a host of techniques from the classical ideas of Euler to the modern ideas of R. W. Gosper, Jr., H Wilf, and D. Zeilberger, of how to simplify finite sums that are likely to appear in the course of one's work. ... This work should prove to be an invaluable aid to students and researchers working in all areas of mathematics. The author's 'hope is to find this summa on your desk—just as Thomas's original was found on the altar!' and the reviewer agrees. -Sills V. Andrew, Mathematiacl Reviews, August 2008
""Summa Summarum is not your typical 'tables of sums' reference book. … this book is not filled with page after page of equations and evaluations. Instead, most of the pages are filled with proofs, and the identities marked as theorems. … As a textbook, Summa Summarum is self-contained and well-organized, providing a solid introduction on how to evaluate finite sums."" -O-Yeat Chan, CMS Notes, November 2008"