Fibonacci and Catalan Numbers: An Introduction
by Ralph Grimaldi
The book is a comprehensive introduction into the subject of the Fibonacci and Catalan numbers and their many properties and uses. It is a tastefully written and well organized textbook that could be used for self study and easy references. The book consists of two parts: the first seventeen chapters cover the Fibonacci and Lucas numbers, followed by 19 chapters that cover the Catalan numbers; both touch on generalizations: the alternate Fibonacci, Narayana, Motzkin, Schröder, and Generalized Catalan numbers.
The two parts have a similar structure. A short historical background, the traditional motivating problems, additional examples that lead to the discussed families of numbers, more theory and more examples, generalizations, a final example, bibliography. The index at the end of the book is common to the two parts as is the collection of solutions to the odd-numbered exercises. (Each chapter ends with a good deal of those.)
The chapters "A Final Example?" come with a question mark. Why? After the many examples where the number families arise, the reader may tend to believe that if a few first terms of a family appear in a certain situation the rest will automatically follow. The two chapters serve a gentle warning that this is not always so. Inductive generalizations are no substitute for proof.
Naturally, there is a plenty of proofs there that explore various techniques: mathematical induction, matrix algebra, recurrence relations, combinatorial arguments. The proofs are truly polished, with no omitted steps (that I could find), and should be easily followed by anybody with undergraduate math proficiency. The book grew out of a respectable number of minicourses presented over a period of 20 years. As the author writes in the preface, "... the presentations were developed so that everyone in the audience would be able to understand at least some, if not a substantial amount, of the material." And later on: "Since the book is to be regarded as an introduction, examples and, especially, proofs are presented with detailed explanations. Such examples and proofs are designed to be careful and thorough."
I believe the author has really achieved those goals. A casual reader - even after quick browsing through the book - will come with a realization of how ubiquitous the Fibonacci and Catalan family of numbers are and how broad are their applications in mathematics and natural sciences. A student who takes the study more seriously will, in addition, learn of their numerous and often beautiful and surprising properties, and master many methods of mathematical proof, even if the book is not read sequentially.