Review of analytic combinatorics in several variables by Robin Pemantle and Mark Wilson

The use univariate generating functions in combinatorial enumeration is a classic topic. In contrast, as the authors explain in the preface, analytic combinatorics in several variables was in its infancy as recently as in the 1990s. It is therefore not surprising that this book is not a textbook, but a collection of research results of the last 15 years. It is certainly a very high-level book, even for the Cambridge Series in Advanced Mathematics, where it is published. Part I, Combinatorial Enumeration, can be viewed as a very high-level introduction to the rest of the book in that it mostly (but not exclusively) deals with univariate generating functions. Readers who have carefully read the books of Enumerative Combinatorics, Volume 2 by Richard Stanley and Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick will find the chapter easier to read than others, but not easy. The authors cover three particularly frequent classes of generating functions. These are, in increasing order of containment, rational functions, algebraic power series, and differentiable finite power series. The latter turn out to be precisely the power series whose coefficients satisfy a linear recurrence relation with polynomial coefficients, such as