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Full Description
Starting from simple generalizations of factorials and binomial coefficients, this book gives a friendly and accessible introduction to $q$-analysis, a subject consisting primarily of identities between certain kinds of series and products. Many applications of these identities to combinatorics and number theory are developed in detail. There are numerous exercises to help students appreciate the beauty and power of the ideas, and the history of the subject is kept consistently in view. The book has few prerequisites beyond calculus. It is well suited to a capstone course, or for self-study in combinatorics or classical analysis. Ph.D. students and research mathematicians will also find it useful as a reference.
Contents
Inversions
$q$-binomial theorems
Partitions I: Elementary theory
Partitions II: Geometry theory
More $q$-identites: Jacobi, Guass, and Heine
Ramanujan's $_1\psi_1$ summation formula
Sums of squares
Ramanujan's congruences
Some combinatorial results
The Rogers-Ramanujan identities I: Schur
The Rogers-Ramanujan identities II: Rogers
The Rogers-Selberg function
Bailey's $_6\psi_6$ sum
A brief guide to notation
Infinite products
Tannery's theorem
Bibliography
Index of names
Index of topics
