Art of Proving Binomial Identities

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.

• 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.

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