Multivariate generating functions

Here are some links to MathSciNet searches:

• anywhere "generating function", "multiv*"
• class 05A15, anywhere "multiv*", "generating function"
• class 05A16
• anywhere "tiling", "generating function"

Here is a personally biased and purposely short list of what I consider essential reading. There are many alternative books, some quite well-known, but I have found the ones listed more helpful.

• A good introduction (albeit with some errors) to generating functions and asymptotic coefficient extraction is contained in Robin Pemantle's lecture notes (Stanford 1999-2000). For those unfamiliar with generating functions, the book generatingfunctionology by Herbert Wilf is recommended. There are many other books covering GFs and their use in combinatorics, one of the best-known being Richard Stanley's two volumes on enumerative combinatorics. The ongoing book project Analytic Combinatorics of Philippe Flajolet and Robert Sedgewick is well worth following. Uses of GFs in probability are found in many standard textbooks.

• For asymptotic enumeration, the original 1974 survey by Edward Bender and the 1995 survey by Andrew Odlyzko are essential.

• Application areas: Phillippe Flajolet and Robert Sedgewick's book An introduction to the Analysis of Algorithms is worthwhile. Readable books on queueing theory are very hard to come by.

• For general asymptotics of integrals Roderick Wong's book is the best overall to learn from. Many other books discuss this material but in less detail. For more elegance and in places more power, try Elias Stein's Harmonic Analysis or the second volume of Singularities of Differentiable Maps by V.I. Arnold et al (these treat only the real phase, Fourier case). Apparently, only Lars Hormander's The Analysis of Linear Partial Differential Operators deals with the complex phase case. These last three books omit many details and assume a fairly high level of sophistication. Integrals where the phase is stationary on a set of positive dimension are covered in very few books, including Wong and Arnold et al.