by James Mc Laughlin
World Scientific, 2018
As is the case with many other topics in pure and applied mathematics, the study of q-series goes back to Euler. While other early results were discovered by Gauss and Cauchy, the first systematic study is due to E. Heine in 1843, who considered q-series as natural analogues of classical hypergeometric series. The theory was then further developed in the early 20th century by the Rev. F.H. Jackson, G.N. Watson, W.N. Bailey, and Lucy Slater, among a few others.
The further development is best summarized by George E. Andrews who in the Foreword to the book under review wrote that “things blossomed in the late 1960s with the discovery that the world of partitions had its natural home in q-series. Seemingly esoteric objects like very well-poised q-hypergeometric series turned out to be the central tools for the exploration of partition identities.
“With the Discoveries of Ramanujan’s Lost Notebook in the 1970s and Bailey Chains in the 1980s, q-series have subsequently become a center of intense research.
“James (Jimmy) Mc Laughlin has gathered together the components that fueled this resurgence of q-series and combined them into a natural and coherent text. The central theme revolves around the Bailey Chain, its extensons and implications. The book concludes with a nice development of related continued fractions and an introduction to Ramanujan’s moch theta functions. There are sufficient exercises included to allow this book to be used as a text in a graduate course.”
An informative review of this book, with references to the earlier liteature, appeared in MathSciNet under MR3752164. The reviewer, David M. Bressoud, ends by writing, “Mc Laughlin has produced an admirable book, clearly and knowledgeably written, upon which one could build a challenging undergraduate seminar as well as a graduate course designed to lead toward today’s research questions.”
The book ends with seven appendices, from “Frequently Used Theorems” to “Selected Summation Formulae”, which are intended for quick reference. The book is available in hardcover, softcover, and as an e-book.