The Power of \(q\): A Personal Journey

The \(q\) of the title is the name of the variable in basic hypergeometric series, that are known for short as \(q\)-series or \(q\)-hypergeometric functions. Because of the structure of the series there are a large number of recursive and reflective relationships among them, and these figure largely in the proofs.

Most of the results deal with partitions of various sorts, not only the familiar representation of a positive integer as a sum of positive integers, but also some generalizations of this, and some specializations such as representing an integer as a sum of two or four squares of integers. The theory originated with Euler and Jacobi (the Jacobi triple product identity is a famous example). Its greatest modern practitioner was Ramanujan, and it is still an active research area today.

The book consists of a large number of results that can be proved using these series. In every case the series are viewed as formal power series (and formal infinite products) rather than analytic functions, and the proofs depend on a great deal of cleverness but not on advanced knowledge. The book starts with a concise introductory chapter that explains all the functions and methods. The rest of the book is a collection of (often short) chapters, each dealing with a particular problem and its solution by these methods. The book is well-written and the logic is easy to follow, although the proofs in some cases are intricate and require making dozens of transformations using the known relations among these functions. Most of the chapters seem to be independent of the others, although related subjects are grouped together. The book is better for browsing and dipping into than for reading straight through.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.