This is an exposition of a group of methods for working with hypergeometric sums, developed roughly from 1945 to 1995. Hypergeometric has a specific meaning here, but you can think of it roughly as any sum or infinite series where the summand is made up of binomial coefficients, factorials, polynomials, and powers. To get a quick overview of the power of these methods, look at the *American Mathematical Monthly* article “How To Do *Monthly* Problems With Your Computer” by Nemes et al. (v. 104, 1997, pp. 505–519), that gives a sample of 27 problems from the *Monthly* Problem Section that can be done mechanically using these methods.

The cryptic title *A=B* refers to the methods’ application to proving hypergeometric identities, but in most cases they can produce a closed form expression for a finite or infinite sum without your needing to guess it first. The book gives a gradual and mostly historical introduction to the algorithms: Sister Celine’s (most of historical interest today, but a good introduction), Gosper’s, Zeilberger’s creative telescoping, Wilf-Zeilberger (WZ), and a more powerful form of WZ called Hyper. The last chapter collects a lot of fragments and ideas for further work. The algorithms are all conceptually simple but messy to work out, so they are ideal for computer implementation, and some of the implementations (mostly in Maple and Mathematica) are discussed here.

The present book is nearly 20 years old. The algorithms given here are essentially the last word for the problems they can solve, so the book is still up-to-date as far as it goes. There have been some new algorithms developed since then. A more recent book (that I have not seen) is Koepf’s *Hypergeometric Summation* (2nd edition, 2014) that covers the same topics as the present book plus the newer algorithms; it is slanted towards implementation while the present book is slanted toward the mathematical theory.

Note that this book is also available online.

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