The Art of Proving Binomial Identities accomplishes two goals: (1) It provides a unified treatment of the binomial coefficients, and (2) Brings together much of the undergraduate mathematics curriculum via one theme (the binomial coefficients). The binomial coefficients arise in a variety of areas of mathematics: combinatorics, of course, but also basic algebra (binomial theorem), infinite series (Newton’s binomial series), differentiation (Leibniz’s generalized product rule), special functions (the beta and gamma functions), probability, statistics, number theory, finite difference calculus, algorithm analysis, and even statistical mechanics.
The book is very suitable for advanced undergraduates or beginning graduate students and includes various exercises asking them to prove identities. Students will find that the text and notes at the end of the chapters encourages them to look at binomial coefficients from different angles. With this learning experience, students will be able to understand binomial coefficients in a new way.
- Provides a unified treatment of many of the techniques for proving binomial coefficient identities.
- Ties together several of the courses in the undergraduate mathematics curriculum via a single theme.
- A textbook for a capstone or senior seminar course in mathematics.
- Contains several results by the author on proof techniques for binomial coefficients that are not well-known.
- Ideal for self-study, it contains a large number of exercises at the end of each chapter, with hints or solutions for every exercise at the end of the book.
Table of Contents
Introducing the Binomial Coefficients. Basic Techniques. Combinatorics. Calculus. Probability. Generating Functions. Recurrence Relations and Finite Differences. Special Numbers. Miscellaneous Techniques. Hints and Solutions to Exercises.
Michael Z. Spivey is Professor of Mathematics at the University of Puget Sound, where he currently serves as chair of the Department of Mathematics and Computer Science. He earned his PhD in operations research from Princeton University. He has authored more than 25 mathematics papers, most of which are on optimization, combinatorics, or the binomial coeffcients.
Many undergraduate mathematics curricula require a course, usually called a capstone or senior seminar, that exposes upper-level students to connections between the canonical subfields studied in their discipline. One topic that can be used to cover many aspects of the undergraduate mathematics curriculum is binomial coefficients. Binomial coefficients are used either implicitly or explicitly in statistics, probability, infinite series, calculus of series, number theory, linear algebra, and of course basic algebra, as seen in the binomial theorem and Pascal’s triangle. They also show up in some special functions such as the beta and gamma functions as well as in special sequences of numbers such as the Fibonacci, Catalan, and Bernoulli numbers. In this text, Spivey (Univ, of Puget Sound) develops the study of binomial coefficients and binomial identities from the definition, and guides the reader through over 300 identities. He also includes over 300 exercises, providing hints and/or solutions to each one. The student reader will gain an appreciation of the numerous connections between these different areas and will also gain confidence in writing proofs. Anyone solving problems found in mathematics journals—or preparing for mathematics contests—will find this text invaluable.
--J. T. Zerger, Catawba College, from CHOICE